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SCHOOL OF 

CIVIL ENGINEERING 

INDIANA 

DEPARTMENT OF TRANSPORTATION 



JOINT HIGHWAY RESEARCH PROJECT 

JHRP-75-8 

Final Report 

COMPUTER ANALYSIS OF GENERAL 
SLOPE STABILITY PROBLEMS 

Ronald A. Siegel 



! 



I 



: 




*s ^ 




UNIVERSITY 



Digitized by the Internet Archive 

in 2011 with funding from 

LYRASIS members and Sloan Foundation; Indiana Department of Transportation 



http://www.archive.org/details/computeranalysisOOsieg 



JOINT HIGHWAY RESEARCH PROJECT 

JHRP-75-8 

Final Report 

COMPUTER ANALYSIS OF GENERAL 
SLOPE STABILITY PROBLEMS 

Ronald A. Siegel 



Final Report 
COMPUTER ANALYSIS OF GENERAL SLOPE STABILITY PROBLEMS 

T0: J- F. McLaughlin, Director June 4, 1975 

Joint Highway Research Project 

fdhm- u i mi u i „ , Project: C-36-36K 

FROM. H. L. Michael, Associate Director 

Joint Highway Research Project File: 6-14-11 



n kl e „ F " al Report "Computer Analysis General Slope Stability 
Problems 1s submitted for acceDtance in fulfillment of the 
objectives of the approved JHRP Research Study titled "A 
Computer Analysis of an Irregular Failure Surface by the Method 
S ii C f S ' The researcn and report were performed by Mr. 
Ronald A. Slegel, Graduate Instructor In Research on our staff, 
under the direction of Professors C. W. Lovell and W. D. Kovacs. 

c + kJf? re P°, rt Includes a review of methods available for slope 
stability analysis, and a description of the computer program 
deve oped The program has great versatility and should be of 
considerable value to the ISHC In economically analyzing cut 
u^fVm^m^" 1 h 10 ???\ Anothe f re P or t. JHRP-75-9, 1s the "STABL 

nlr*" * .t^ w111 be useful to an * one who uses the computer 

programs of this research. 

The Report 1s submitted for approval and acceptance. 

Respectfully submitted, 



Harold L. Michael 
Associate Director 



HLMrmf 



cc: W. L. Dolch m. L. Hayes C 



F. Scholer 



!• \' ^* ew c - W. Lovell m. B. Scott 

S- I- ^ b * on G - W. Marks K. C. S1nha 

M l* f^n J' F ' Marsh H. R. J. Walsh 

M. J. Gutzwlller R. D. Miles L r 

G. K. Hallock G. T. Satterly E .' 



E. Wood 
J. Yoder 
S. R. Yoder 



Final Report 
COMPUTER ANALYSIS OF GENERAL SLOPE STABILITY PROBLEMS 

by 

Ronald A. Sieqel 
Graduate Instructor In Research 



Joint Highway Research Project 
Project No. : C-36-36K 
File No. : 6-14-1 1 



Prepared as Part of an Investigation 

Conducted by 

Joint Highway Research Project 

Engineering Experiment Station 

Purdue University 

in cooperation with the 

Indiana State Highway Commission 



Purdue University 
West Lafayette, Indiana 
June 4, 1975 



11 



ACKNOWLEDGMENTS 



The author wishes to express his gratitutde to Dr. 
W. D. Kovacs, Assistant Professor of Civil Engineering, 
and Dr. C. W. Lovell, Professor of Civil Engineering at 
Purdue University, for their valued assistance and guidance 
during the course of the project. 

The project was funded by the Joint Highway Research 
Project, Engineering Experiment Station, Purdue University, 
Dr. J. F. McLaughlin, Director, in cooperation with the 
Indiana State Highway Commission. 

Special thanks are due to Mr. W. J. Sisiliano, Soils 
Engineer, Indiana State Highway Commission, for his as- 
sistance in defining the capabilities required of the 
computer program developed, and to Mr. John Bellinger of 
the Indiana State Highway Commission Computer Center for 
his assistance in establishing the program on line at that 



center, 



Finally, th e author wrshes to thank Mr. Yogesh D. Shah, 
who drafted the figures, Miss danioe Wait for her efforts in 
typing the draft, and Mrs. Paul w. Velten who has typed 
the final manuscript. 



Ill 



TABLE OF CONTENTS 

Page 

LIST OF TABLES. 

v 

LIST OF FIGURES 

vi 

LIST OF SYMBOLS 

viii 

ABSTRACT. 

» xiii 

INTRODUCTION 

1 

METHODS AVAILABLE FOR SLOPE STABILITY ANALYSIS 4 

* • • • T 

Limiting Equilibrium Methods 

t™** 1 ?? E ^ uilibri ^ and Fundamental Assumptions 6 

Logarithmic Spiral Shear Surface Prions 6 

= Case ... iq 

Friction - Circle' Method ^1 

Method of Slices . 18 

Ordinary Method of' slices " \\ 

Simplified Bishop Method \i 

Bishop's Method, Rigorous W\ 

Janbu's Generalized Procedure of' slices ' ' 39 

Morgenstern and Price Method " * AA 

Spencer's Method 



Carter's Method 



49 

METHOD CHOSEN FOR COMPUTER PROGRAM 57 

PROGRAM STABL 

64 

Division of Mass into Slices above the 
Assumed Shear Surface 

Hill TJrtl Calculaw ™ °^ch sue; : : ; : ; t\ 

70 

SEARCHING TECHNIQUES- . . . 

75 

Random Numbers 

Circular Surface Generation-' '.'.'. l^ 

Sliding Block Surface Generation .' .' ." .' .' [ [ [ \\ 



IV 



SS2SS°2 2 F CARTER ' S METHODWITH SPENCER'S METHOD ^^ 

ASSUMING PARALLEL SIDE FORCES 102 

SUMMARY AND RECOMMENDATIONS FOR FUTURE WORK 108 

LIST OF REFERENCES. . . 

110 

APPENDIX A: LISTING OF PROGRAM STABL 113 

STABL Main Program 

Subroutine READER 

Subroutine QUIT. ..." 117 

Subroutine PROFIL. .....' 121 

Subroutine ANISO ...'."] 122 

Subroutine SURFAC. . 131 

Subroutine WATER ......* 135 

Subroutine LOADS ...*.'.* 138 

Subroutine EQUAKE. ..'..' 141 

Subroutine LIMITS. ..'.'! 145 

Subroutine EXECUT. . 146 

Subroutine SLICES. ..'.']." 149 

Subroutine INTSCT. . . ^^^ 

Subroutine WEIGHT. . 156 

Subroutine SOILWT. . 158 

Subroutine FACTR ....*,' 163 

Subroutine RANDOM. . .* ] [ 165 

Subroutine RANSUF. ..."." 170 

Subroutine SORT. ...'." 184 

Subroutine SCALER. ..*.'.'.' 197 

Subroutine PLOTIN. . 198 

Subroutine PLTN. ...'.' 200 

Subroutine PSTN. 205 

210 



LIST OF TABLES 
Table 



Page 



1. Equations and Unknowns Associated with Procedure 
of Slices _. 

24 

2. Factors of Safety Calculated Using Carter's 
Method • 104 



VI 



LIST OF FIGURES 

Figure _, ■ 

Page 

1. Basic Forces Associated With Limiting 
Equilibrium 7 

2. Planar Shear Surface 13 

3. Logarithmic Spiral Shear Surface and Force 
Orientation 16 

4. Mass Above Assumed Shear Surface Divided into 
Slices and Linearization of the Slices' Bases . 21 

5. Basic Forces Acting upon Individual Slice ... 23 

6. Forces Considered for Ordinary Method of Slices 25 

7. Boundary Forces Acting on the End Slices. ... 33 

8. Lever Arms of Nonveiller Extension of Bishop's 
Approach 38 

9. Infinitesimal Slice and Force System 40 

10. Slice and Resultant of the Side Forces. ... 50 

11. Slice Forces Referenced to an Arbitrary Point . 54 

12. Forces Considered for Computer Model 61 

13. Locations of Slice Divisions 65 

14. Geometry of Slices 67 

15. Heterogeneous Slices Partially Submerged. ... 68 

16. Water Forces Acting Upon Submerged Slices ... 71 

17. Approximation of Head for Pore Pressure 
Calculation 73 

18. Selecting the Initial Line Segment 78 

19. Probability of the Location of the Initial Line 
Segment 80 



Vll 

Fi ? ure Page 

20. Family of Circles Having Initial Line Segment 

as a Chord 2 2 

21. Back-Calculating a Deflection Limit for Circular 
Surface 83 

22. Limitations Imposed on Deflection Angle .... 85 

23. Limitations Imposed on Inclination of Final 

Line Segment 87 

24. Deflection Limits for Successive Line Segments 

of Irregular Surface 90 

25. 100 Generated Circular-Shaped Surfaces 93 

26. 10 Most Critical Surfaces of Those in Figure 25 95 

27. Simple Sliding Block Problem 97 



28. Sliding Block Generator Using More than Two 
Boxes .... 



99 



29. Generation of Active and Passive Portions of 
Sliding Surface 10 

30. Comparison of Carter's Method to Spencer's 

Method 105 

31. Curvature of Slope Composed of c' - <j> • Soil . . 107 



Vlli 



a 

A 



B 



c' 



LIST OF SYMP30LS 



- point of application of n' measured from 
the left side of slice. 

= y - h 

= slope of slice base. 



b l = height of left side of slice. 
b 2 = he ight of right side of slice. 



- intercept of slice base with respect to 
coordinate system. 



a - effective stress Mohr-Coulomb strength 
intercept available. 

cjc = peak effective stress Mohr-Coulomb strength 
intercept. 

c m ~ mobilized "cohesion." 

C a = component of the available shear resistance 
along an assumed shear surface which is 
independent of N'. 

AC; = available shear resistance at base of slice 
wnich is independent of N' . 

E' 

AE 



E' 
o 



E' 
n 



f (x) 



- internal effective interslice normal force. 

(dE«) = change of E' across width of finite (in- 
finitesimal) slice. 

= J^!^ 1 ^ horizon tal force acting on the left 
of the first slice. 

= of^h^T 6 bor t Zontal force act ing on the right 
or tne last slice. 

= distributional relationship assumed for the 
side force inclinations. 

= factor of safety (strength reduction factor) . 



IX 



h 
h 



« height of slice measured at centerline 

= SSL°f ^Jr -^tionship for „„ 

- horizontal earthquake coefficient. 

k v - vertical earthquake coefficient. 

K 

~ " A k (tan <f> • / F - a) 

L , 

- ~ A m (tan <j> ' / f - a} + i 

' withrnTsxLe*. Unear » ia "o-»ip fo'r f (x) 
= sL^surr™!" Withi " ™ aES ab °« — -»* 
= SaTa^ce" S " d ° f -« above assu.ee 

- P (tan 0^ / f - a) 
■ Point about which moments are su^ed. 

= Si3*.2fvi 1 3 a ^JS; t iS hlp of the first 

within a slice 9 th res P ect to position 
= (1 + A 2 ) c . / F + q (tan . / F - a) 

within a slice h res P ec t to position 

" rSSUltant of si de forces of a slice. 



k 



k h 



M 

n 



n 

N 
N' 




P 



r 
n 

r 
o 

r 
s 

r 
u 



* E"S2iS "^circle. ° f ' ^^rittaic spiral, 

- lever arm of a<J about an arbitrary point , 0) . 

- lever arm of «. about an arbitrary point (OK 

- reference radius of logarithmic spiral. 

- lever ar m of S r about an arbitrary point ,0, . 

- pore pressure coefficient (H_) 

- lever arm to O of a h„„ . 
? r random numLfo? ra'nge* otV --nt centers- 
force acting upon loa.Hfn!- ° r rand om 
surface. P logarithmic spiral shear 

= resultant of driving forces acf.n„ 

above the assumed shea? surface 9 " mass 

= c^uld'bf mobiliLd^:! 1 ^^ 16 Shear Str -^h which 
surface. ° ng an assumed shear 

- available shear resistance at the base of a 
= SSS^ ^s^/--/---- mobile 

"* ^ = ^ili^iuf aT^L^f ^.^li-itin, 
(infinitesim.n 2?> se of a fln ^e 



R d 



S 
a 



AS 



S 

r 



T 

AU 
a 

AW (dW) 



Ax (dx) 



(infinitesimal) slice. 

- length of shear surface li ne se g me nts. 

- water force acting on base of slice. 

- water force acting upon top of slice. 
= total weight of a slice. 

= slice^^-a ^enTcenter™ "» ~*« - 
- «l«h of f lni te (infin.tesi.al, slice. 



XI 



X 

n 

x 

s 



X 

w 



X 



= lever arm of AN' about an arbitrary poinr (0) . 
= lever arm of AS r about an arbitrary point (0) . 
= lever arm of AW about an arbitrary point (0) . 
= internal interslice shear force. 



AX (dX) = change of X across width of a finite 
(infinitesimal) slice. 



X o 



X 

n 



y 



= slice? 31 f ° rCe aCtlng ° n the left of the ^rst 

= s!ice? al f ° rCe aCtlng ° n the ri ^ ht of the last 

= distance in y direction from the center of a 
slice base to the moment center. 

Y = office? ° f the ShSar SUrface at left Bide 

AY (dy) = change of elevation of the shear surface across 
finite (infinitesimal) slice. across 

^ = lefTslde 1 of SLe!^ ° f •"~ tl ™ ^^ « 

Ay; (dy^) = change of elevation of the line of effective 
thrust across finite (inf initesimf iffSe 

a = inclination of slice base. 

a dc = angle between R^ and C 

d a 

a dn = ar >gle between R^ and N" 

d 

a ds = angle between R, and s 

d r 

Ql = fo^initiai ?f n f° unt -clockwise direction limit 
ror initial line segment of shear surface 



Xll 



&xupe stability analysis. 

e 2 = inclination of counterclockwise directior limit 
surface? ^^ ° f ^galar-shaped shTar 

62 = l?ni inati ° n ° f clockw ^e direction limit for 

line segment of .rregular-shaped shear surface. 

= Hu.ttSfS^? center of circle * 
~~ SSffLS^.Sf 160 ' 1011 angle for ci - ul - 

1 = counterclockwise direction limif „# i ■ 
of irregu ar-shapod shear subtle ^ S6gment 

2 = irregular fhfTT Umit ° f iine s ^e„t of 
irregular-shaped shear surface. 

mln = cJrcular^h^^ 10 " U,nit f ° r line ^g.ent of 

circuiar-r.haped shear surface. 

= maximum deflection iimi+- -p -. ■ 

circular-.hape^sTearsurSoe' 1 " 6 "^ ° f 

stability analysis. ° PriCe P r °«*>« for slope 
= SJ^i^iSbSr **r-CoulCb shear strength 

** = !ngle?" eCtiVe " re " ^-Coulomb strength 

*m = mobilized strength angle. 

°n = effective normal stress. 

" appoint ShSar StrSn9th " hich «"» ^ mobilized 
" a'rverg^cf 5 ^^^." P01 " °" f *""« surface 



A6 
A6* 

A6 

A6 
A6 
A6 



max 



♦a 



Xlll 



ABSTRACT 

Siegel, Ronald A., MSCE, Purdue University, May _975. 
Computer Analysis of General Slope Stability Problems. 
Major Professor: C. W. Lovell. 

A computer program, named STABL, for performing limiting 
equilibrium analyses by the method of slices for general 
slope stability problems has been developed. A discussion 
of previously available theories and programs precedes the 
presentation of the model of stability analysis used in this 
research and its computer simulation. 

The program can handle slope profiles having multiple 
slope ground surfaces. Any arrangement of subsurface soil 
types having different so^l properties can be specified. 
Pore pressures may be related to a steady state flow domain, 
related to the overburden, or specified within zones. Sur- 
charge boundary loads and pseudo-static earthquake loadings 
are also considered. 

Three trial shear surface generators have been developed 
to generate surfaces either of circular shape, of general 
irregular shape, or of specified character such as sliding 
blocks. Restrictions may be imposed to generate only 
surfaces of interest. 

Graphical output provides means for checking input and 
for quick evaluation cf results. Print character plots are 



XIV 



processed on line printers or teletype terminals. Plotting 
devices provides plots of higher resolution. 

The program has been written in Fortran IV source 
language, and is routinely used on Purdue University's 
Computing Center's Control Data Corporation 6500 computer. 
With full plotting capabilities 66K octal of core storage 
is required for execution. Program length is 6000+ state- 
ments including components. Another version is being modi- 
fied for use at the Indiana State Highway Commission's 
Computer Center's IBM 36 facility. 



INTRODUCTION 



Wherever there is a difference in the elevation of the 
earth's surface, whether it be the result of man's actions 
or natural processes, there are forces which act to restore 
the earth to a level surface. The process in general is 
referred to as mass movement. a particular event of special 
interest to the geotechnical engineer is the landslide. The 
geotechnical engineer is often given the task of ensuring 
the safety of human lives and property from the destruction 
which landslides can cause. 

In order to accomplish this task he must have a 
thorough understanding of the mechanics of the failure 
mechanisms involved. He must be able to reasonably as- 
certain the insitu shearing stress and resistance of the 
materials involved, and if water is present, he must be able 
to define the flow regime and the water's relationship to 
strength. Knowing these, he can formulate a model which can 
be used to make a prediction as to whether a slope will 
perform safely or not. 

The primary cause of motion is a component of gravity 
acting on the sliding mass in the direction of movement 
along a failure surface, other forces may also act. Seep- 
age forces, which result from energy lost by ground water 
flowing through permeable earth, can be and often are 



significant. In regions of seismic activity, earthquake 
displacements can impart significant forces which can initi- 
ate mass movement in an earth slope. 

These forces produce shear stresses within the slope. 
Unless the strength of the earth material is sufficient 
along any possible shear surface, a movement of mass down 
slope will occur. Except for a few special cases, the 
position and shape of the surface most likely to fail or 
most likely to approach failure is not obvious by exami- 
nation of slope profiles. Therefore an analysis using 
limiting equilibrium usually requires that a number of trial 
surfaces be evaluated in order to determine the position and 
shape of the most likely path of failure. 

The method of slices, particularly applicable to 
heterogeneous soil systems, requires a great amount of 
computation per trial surface, regardless of the assumptions 
made for analyses. If performed by hand, time in general 
will allow only a few trial surfaces to be analyzed economi- 
cally. However, with the aid of a programmed computer, 
hundreds of trial surfaces can be analyzed quickly and 
economically. 

In highway design, it is common to be concerned with a 
large number of cuts and embankments, each often having 
characteristics sufficiently unique to require individual 
analysis. A civil engineering project such as a highway, 
having great length, invariably encounters geologic 



conditions which vary considerably from location to location 
along its route, e.g., groundwater conditions, surface 
drainage, material properties, and economical construction 
techniques. The need to analyse an adequate number of typi- 
cal slope profiles, in a routine and efficient manner, re- 
quires implementation of a computer programmed to handle 
general problems. 

The primary purpose of this thesis was to develop a 
computer program which will enable a geotechnical engineer 
with an understanding of slope stability to define a 
problem and then be provided some measure of its stability, 
a factor of safety. The program named STABL is a model 
with flexibility able to handle most stability problems, 
without requiring gross simplification to render them 
tractable for solution. 

A number of computer programs were available at the 
time of this undertaking. Their capabilities and limi- 
tations vary widely, as well as their requirements for in- 
put data. It is difficult to assess their capabilities be- 
cause many are still being developed, modified and improved 
by their authors or successors of their authors. Therefore, 
this author does not wish to generalize about their capa- 
bilities. However, it was felt that a new program, ex- 
panding beyond the capabilities of ev i of ,- w 

a^auinues or existing programs, with 

few limitations, and with an emphasis on ease of use was 
needed. 



METHODS AVAILABLE FOR SLOPE STABILITY ANALYSIS 

There are basically two deterministic approaches to 
slope stability analysis. The first concerns limiting eguilL 
brium analyses where strain considerations are of no conse- 
quence; and the second, elastic methods where strain and its 
relationship to stress is of importance. 

The latter can be more realistic for modeling the 
mechanics of slope behavior. However, our xnability to ade- 
quately ascertain the insitu nonlinear stress-straxn be- 
havior of materials composing slopes, severely li mits the 
application of these methods. More often than not, an ade- 
quate assessment of a slope's stability can be made using a 
cruder limiting equilibrium model where the uncertaxnty of 
material properties is confined to the shear strength parame- 
ters selected. 

Methods using theories of elasticity are either exact 
or approximate. Exact solutions have been determined for 
simple cases only. For example the Boussinesq solution 
(1855) determines the distribution of stresses within a 
linear elastic half-space induced by a concentrated force 
Placed upon its surface. A problem somewhat more compli- 
cated has been solved for the stress distribution within the 
body of a slope having a cylindrical face (Roman! , Lovell 
and Harr, 1972, . Exact solutions are not available for 



application to general slope stability problems encountered 
in engineering practice. 

Numerical approximations such as finite difference 
methods and finite element methods offer greater flexi- 
bility for handling complicated problems, including treat- 
ment of nonlinear stress strain relationships, but require 
a great amount of attention to detail, and often the models 
become too complicated for practical use in engineering 
problems. Practical engineering applications require 
simplifications, and when they are introduced, inaccurate 
results may be obtained (Hansen, 1953). 

Limiting E quilibrium Methods 
Limiting equilibrium methods follow one of two ap- 
proaches (Harr, 1966). The first, originating with Rankine 
(1857) , assumes full mobilizatlon Qf the avaUable shear 

strength at all points within a slope. The methods uti- 
lizing this approach are mathematical treatments, in closed 
form, solving statically determinate systems of equations 
satisfying given boundary conditions. The problems that 
have been solved by this approach are limited, due to 
analytical difficulties encountered while arriving at 
solutions. 

The second approach, originating earlier with Coulomb 
(1776), requires assessment of the stability of the material 
above an assumed shear surface, a number of assU med shear 
surfaces are analysed to determine the critical shear 



surface. Equilibrium of the material above each shear 
surface is examined by assuming enough shear strength is 
mobilized to maintain the slope at the verge of failure. 
The factor of safety with respect to failure along a shear 
surface being investigated is usually defined as the ratio 
of the available shear strength to the shear strength 
mobilized. 

Limiting Equilibrium and Fundamental Assumption. 
In general the limiting equilibrium approach is stati- 
cally indeterminate. To illustrate this, a discussion, of 
what is known and not known relevant to a general slope 
stability problem, is presented. Figure 1 may be referred 
to while following this discussion. 

The resultant of the drxving forces, R fl , and its line 
of action can be readily calculated. The driving forces in- 
clude the total weight of all the mass located above the 
assumed shear surface, earthquake forces which may be im- 
parted to this mass, buoyant forces if the mass is partially 
or totally submerged, and boundary surcharge loads which may 
be located upon the ground surface within the extent of the 
assumed shear surface. 

The distribution of effeotive normal stress, „ hlch is 
unknown, can be represented by a resultant effective normal 
force, N-. However, its magnitude and line of action are 
known. The resisting shear strength mobilized along rhe 



un- 





(b) 




FIGURE I - BASIC FORCES ASSOCIATED WITH LIMITING 
EQUILIBRIUM. 



assumed shear surface can also be replaced by a single un- 
known resisting shear force, S r , whose line of action is also 
unknown. 

The shear strength mobilized is that required to main- 
tain the mass, located above the assumed shear surface, in 
a state of limiting equilibrium. If the slope is stable, 
the shear strength mobilized is only a portion of what is 
available. 

A factor of safety may be defined as the ratio of 
available shear strength to the shear strength mobilized to 
maintain limiting equilibrium. in terms of forces, 

F = S a / S r U> 

where S a is the resultant shear force of all the shear 
strength available along the assumed shear surface. The 
resultant shear force required for limiting equilibrium then 
becomes , 

S r = S a / F (2 ) 

where the factor of safety is a strength reduction factor. 

The value of the factor of safety, as expressed, is an 
average value. It is implied that the shear strength mobi- 
lized along the shear surface is proportional to the avail- 
able shear strength. The limiting equilibrium model does 
not account for stress concentrations. 



Having defined the factor of safety, the problem be- 
comes a matter of determining a unique value for it. 
Relationships between the forces involved must be estab- 
lished. The first of these relationships may be determined 
by satisfying static equilibrium. 

If moment equilibrium of the free body isolated by the 
assumed shear surface is to be satisfied, the three re- 
sultant forces shown in Figure 1(a) must not produce a net 
moment about any point. Below is an expression for moment 
equilibrium about an arbitrary point 0, 

R, r,-N'r-Sr=0 (3) 
d d n r s 

where r, , r , and r are lever arms about point to the re- 
al n s 

sultant forces R,, N' , and S , respectively. Since the 

lines of action of N* and S are unknown, r and r are 

r n s 

also unknown. 

If force equilibrium is to be satisfied for the same 
free body, the summation of forces in two mutually exclusive 
directions must not result in a force imbalance. Expressions 
for force equilibrium in directions parallel and perpendi- 
cular to the resultant driving force, R, , are as follow, 

R d - S r cos a ds - N' cos a dn = (4a) 

S,. sin a, - N' sin a, = (4b) 
r as an 



where a ds and a, are the angular differences of inclinati 



on 



(Figure 1(b)) between the resultant driving force R, and the 



10 



resultant forces S and N', respectively. The angles a-, 
and a , are unknown, because of no information with regard 

to the lines of action of S and N'. 

r 

An audit of the unknowns and the independent equations 
available to relate them, when the free body is at limiting 
equilibrium, shows a total of seven unknowns; F, S , N', 
r , r , a , , a , , but only three equations from which a 
solution for a unique value of the factor of safety is de- 
sired. The problem is clearly statically indeterminate. 

In order to establish static determinancy, assumptions 
are required regarding the forces involved, or stress-strain 
considerations must be introduced. Wishing to avoid the 
complications and uncertainties involved with the latter, 
assumptions are made regarding the unknowns. 

The basic assumption, made with regard to limiting 
equilibrium methods to be discussed, is that the Mohr- 
Coulomb failure criteria is valid. A linear relationship 
between shear strength and effective normal stress is 
commonly used. 

T f - c' f + a^ tan $£ (5) 

where t f is the shear strength on the failure surface at the 

verge of failure, c£ and <f>' are effective stress parameters 

describing the intercept and angle of inclination of the 

Mohr-Coulomb failure envelope, and a' is the effective 

n 

normal stress on the failure surface at failure. 



11 



Other criteria may be used instead of peak stress 
failure, such as residual shear strength, for example. 
Therefore expression (5) is rewritten as, 



T a = c a + °n tan *a (5a) 



where T is the available shear strength, c' and <}> ' are 

a a 

effective stress parameters describing the intercept and 
angle of inclination of the Mohr-Coulomb envelope for some 
criteria other than peak stress failure. 

A nonlinear assumption could be made for the relation- 
ship between shear strength and effective normal stress. 
However, such an assumption will not be presented here. 

The relationship between available shear strength and 
effective normal stress can be expressed in terms of forces, 

S a = c; + N' tan ^ ( 6 ) 

where C^ is the resultant of the component of available 
shear strength which is independent of the normal stress 
distribution. The resisting shear force required to main- 
tain limiting equilibrium is then, 

C* + N' tan *' 
a r a 

s r = (7) 

F 

By examining the two components of the resisting shear 
force, c; / F and (N« tan <^) / f, two observations can be 
made (Figure 1(c)). First, the magnitude of the component 



12 



c; / F is dependent only upon the value of the factor of 
safety, and its line of action is dependent only open the 
geometry of the assumed shear surface; ^ (Figure 1(c)) ^ 
*no„n. Although its magnitude will vary dependlng UDon ^ 
value of factor of safety, its direction is fixed. 

The second observation which can be ma de, is that the 
component of shear resistance assumed dependent upon the 
normal stress distribution „ust act perpendicular to the 
resultant effective normal force, „■ . These tuo obser . 
vations do not add any si g „if ioant infomatio „ ^ uin 
assist in arriving at a solutron for the factor of safety, 

F, for a shear surface of oenprsi ck-, 

general shape. However they do, 

if the assumed shear surface is a plane. 

For a planar shear surface th*> fr^f 

ce ' the fictional and cohesive 

exponents of the resisting shear force, S,, act in the sa.e 
auction, along the plane of the shear surface (Pigure 2(a) 
and2 ' b,K ^^^dc-^.Prg^,^^,^,^ 
egual. The resultant effective normal f orce , „., a „ s per . 
Pedicular to the planar shear surface. Healing this, 

°dn (»*>*» 2(b)) beco.es a *no„n which can be calculated. 
By su^ing forces paraUel and ^^^ ^ the pian ^ 

shear surface, the following two eguations for force eguili- 
bnum are obtained. 

S r - R d sin *dn = (8a) 

»' - R, cos a dn . o (8bJ 



13 




FIGURE 2 - PLANAR SHEAR SURFACE. 



14 



The solution becomes statically determinate for a 
Planar shear surface. The resisting shear force, S r , and 
the effective normal force, N', can be solved for by use of 
the two independent equations of force equilibrium (8). 
Having calculated the effective normal force, N', the 
available shear resistance, s^ can be calculated with 
expression (6), and finally the factor of safety, F, can 
be calculated using expression (1) . 

For shear surfaces of general shape, the problem re- 
mains statically indeterminate. if a solution is to be 
arrived at, a distribution of effective normal stress along 
the assumed shear surface must be determined or assumed, so 
that the magnitude and line of action may be established 
for the resultant effective normal force, N'. 

If this is done, N', a dn , and r n will become knowns. 
Thus the unknowns are reduced to four. with the three 
equilibrium equations, plus the assumed shear strength 
relationship with normal stress, a solution can be arrxved 
at. The value determined for the factor of safety is depen- 
dent upon what assumption is used with regard to the ef- 
fective normal stress distribution. 



Logarithmic Spiral Shear Surf 



ace 



The limiting equilibrium approach originating with 
Coulomb is a statically indeterminate one. except when the 
shear surface assumed is a plane, as previously mentioned, 



15 



or a logarithmic spiral (Froiilich, 1953) of the form 

*- - >- « tan 4> ' 

r = r o e v m (9) 

where r is the radial distance from the center of the spiral 
to a point on the spiral, r Q is a reference radius, 6 is the 
angle between the reference radius and a radial line to the 
point on the spiral, and ^ is the effective mobilized 
friction angle of the material composing the slope. A plane 
can be considered a special case of expression (9) , where r 
is of infinite magnitude. 

The shear surface is dependent upon the mobilized 
friction angle which is unknown. Therefore the procedure re- 
quires that the spiral's center and reference radius be fixed 
while determining the effective mobilized friction angle. 
Given a spiral center point and reference radius, an effective 
mobilized friction angle is assumed. 

When moments are summed about the center of the spiral, 
the moment contributed by the resultant of the effective 
normal stresses, N', and the resultant of the frictional 

components of the mobilized shear strength, N' tan A ' , cancel 

Y m 

each other. The resultant of these two forces, R, acts 
through the center of the spiral (Figure 3). Therefore by sum- 
mation of moments about the spiral's center, equilibrium can 
be examined without requiring knowledge of the normal stress 
distribution along the shear surface. The average component 
of shearing resistance due to "cohesion" required to 



16 




17 



maintain equilibrium of the slope along the assumed shear 
surface can then be assessed by satisfying moment equilibrium 
about the center of the sprial. 

The factor of safety, F, is defined as the ratio of 
the "cohesive" shear strength available, c\ to the 
"cohesive" shear strength mobilized, c ;. This calculated 
factor of safety must be equivalent to that defined by the 
ratio of the available frictional shear strength, tan * • , 

to the frictional shear strength mobilized, tan d> ' 

r m* 

c' tan <t> ' 
F s a _ y a 

el tan rf,' (10) 

m v m 



If the above relationship is not satisfied, other 
values of ^ must be assumed, repeating the procedure, trial 
and error, until it is satisfied. To determine the critical 
shear surface, a number of spiral centers and reference 
radii should be evaluated. 

<t> = Case 

The shearing resistance may be written as a value inde- 
pendent of the state of stress, rather than in terms of 
o; and *;. This is equivalent to stating that the value of 
♦; in equation (5) is zero. For this case the radius of the 
spiral becomes a constant, i.e., the spiral degenerates into 
a circle. when the shear strength of a slope is so assumed 
to be independent of the normal stresses acting upon the 
shear surface, and these stresses all act toward the center 
of the circle, their contribution to the moment about the 



18 



center of the circle is eliminated. Statical determmacy 
with respect to moment equilibrium is achieved, and the 
average shear strength required for equilibrium can be 
calculated from the moment equilibrium equation. The factor 
of safety for the = condition, assuming a circular 
surface, is defined as the ratio of the available shear 
strength to the shear strength mobilized. 

Surfaces, other than logarithmic spirals of the form 
discussed, planar surfaces, and circular surfaces when the 
frictional component of shear strength is not considered 
require assumptions to determine the normal stress distri- 
bution along the shear surface, a number of methods have 
been developed. 

Friction-Circle Method 
The friction-circle method provides a value of the 
factor of safety for any reasonable assumption for the 

distribution of the normal stresses n„ m ^- 

stresses. By making an assumption 

equivalent to assuming that all the normal stress is concen- 
trated at a single point on the failure surface, a lower 
bound value can be established for the factor of safety 
(Frochlich, 1955; Whitman and Moore, 1963). By assuming all 
the normal stress to be concentrated at the two end points 
of the trial shear surface (Frochlich, 1955; Whitman and 
Moore, 1963) an upper bound value for the factor of safety 
can also be established. 



19 

These two cases are extremes which are physically un- 
reasonable. However, the value of the factor of safety 
resulting from the assumption of any reasonable normal 
stress distribution will lie between these two bounds. 
Values of factors of safety calculated by other methods may 
be judged as being reasonable, by comparison with those ob- 
tained by the lower and upper bound solutions. 

A sinusoidal normal stress distribution is often used 
in practice for the friction-circle method, because it is 
intuitively more reasonable (Taylor, 1937). The upper 
bound solution has greater influence on the value of the 
factor of safety than the sinusoidal distribution when 
comparing both to the lower bound solution (Taylor, 19 37; 
Wright, 1969). This may suggest using the value determined 
by the lower bound solution as a conservative value. How- 
ever, the difference, in the values of the factor of safety 
for the lower bound solutions and that for solutions using 
the more reasonable assumption of a sinusoidal normal stress 
distribution, varies. The difference increases with an in- 
crease of the subtended angle of the circular arc defining 
the shear surface. The lower bound 'solution may therefore 
be too conservative for economical design purposes. 

Method of Slices 

The most commonly used procedure of slope stability 
analysis are methods which divide the mass above an assumed 



20 

shear surface into slices (Figure 4). These methods can 
be conveniently applied to problems dealing with hetero- 
geneous slope profiles. The mass is divided so that the 
base of each slice is characterized by a single set of shear 
strength parameters. The shear strength at the base of each 
slice can therefore be described independently of the other 
slices. 

Additional divisions of the mass above the assumed 
shear surface are made to simplify calculations. It is 
convenient to assume that the base of each slice is the chord 
of the curved shear surface, rather than the curved shear 
surface itself (Figure 4). The weight calculations are thus 
greatly simplified. Also the inclination of the chord may be 
used as the average inclination of the actual curved base of 
each slice. Both these simplifications introduce an error 
which can be minimized if an adequate number of slices is 
used. The net error summed for all slices decreases as the 
number of slices increases, providing an adequate number of 
significant figures are used for the calculations. 

The mathematical models presented initially will ignore 
consideration of forces frequently encountered in practical 
slope stability problems such as those produced by buoyancy 
and seepage, earthquake displacements, and ground surface 
boundary loads. Introduction of these considerations do not 
add to the statical indeterminacy of the procedures to be 
discussed, but simply complicate the expressions derived. 



21 




o 

\- 

2 

Q 
LU 
Q 

> 

5 

LU 

(J 

s . 

< 

or cd 

-7-LU 

-J 
Q(0 

<o 

IU2 

>Q 

Sb 

m< 

<N 

q: 
</>< 
cnw 

2D 

i 

UJ 

S2 



22 



This being the case, only the fundamental forces involved 
will be considered at this time. 

Each slice is acted upon by a system of forces as 
shown in Figure 5. Table 1 lists the equations available 
for analysing the stability of a slope by the method of 
slices procedure, as well as the unknowns involved. it is 
seen, that unless only one slice is used, the problem is 
statically indeterminate requiring some simplifying as- 
sumptions to be made. 

Ordinary Method of Slices 

When considering overall moment equilibrium of the mass 
above a circular shear surface, the need to know the po- 
sition of the effective normal force acting on the base of 
each slice, AN*, is not required when moments are summed 
about the center of the assumed circular shear surface. All 
the effective normal stress will act through the circle's 
center, producing no moment about that particular point. 

The overall moment equilibrium equation, taking moments 
about the center of the circle, (Figure 6), becomes 

n n 

I AW r sin a - £ AS r = (11) 

in which AW is the weight of an individual slice, r is the 
radius of the circle, a is the inclination of the base of 
the slice, AS r is the resisting shear force acting along the 
base of the slice, and n is the number of slices. The side 



23 



L 




Line of Effective 
Thrust, y/Oc) 



Assumed Shear 
Surfoce,y(x) 



EVAE' 



(y;+Ay f 'My«.Ay) 



FIGURE 5 - BASIC FORCES ACTING UPON INDIVIDUAL 

9UUL 



03 



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rsma 




FIGURE 6- FORCES CONSIDERED FOR ORDINARY 
METHOD OF SLICES. 



26 



forces are internal forces that do not contribute to the 
overall external moment. 

An error has been introduced into the above expression. 
The moment arm to the weight force, AW , of each slice, 
r sin a, implies that the weight of each slice acts at the 
center of each slice. This is generally not the case. The 
error can be eliminated by using the appropriate moment arm 
to the center of gravity of each slice. If the simplifi- 
cation r sin a is used for the moment arm, the magnitude of 
the error introduced will decrease as the number of slices 
used increases. 

The resisting shear force, AS r , acting along the base 
of each slice is defined as the sum of the cohesive and 
frictional strength available at the base of a slice, 
divided by a factor of safety 

AC + AN' tan A' 
AS r - -» . a (12) 

where AN' is the resultant of the effective normal stress 
acting on the base of the slice, and AC is that component 
of the resultant of the shear strength available at the base 
of the slice, which is independent of the state of effective 
stress. 

By substitution of the expression for shearing resis- 
tance (12) into the expression for overall moment equilibrium 
(11) and solving for the factor of safety, the following ex- 
pression is obtained, 



27 



n 



I (AC' + AN' tan $ ' ) 

i a a 



F n " (13) 

I AW sin a 

1 



Not needing to know the location of the effective 
normal force acting on the base of each slice, the number of 
unknowns reduces to 4n-2. The 3n equations of equilibrium 
are still not adequate for statical determinacy, and addi- 
tional assumptions are required. These final assumptions 
are generally made with respect to the side forces. 

If forces are summed normal to the base of the slice 
shown in Figure 5 it will be observed that the only unknowns 
which would contribute to a resulting expression would be 
the interslice side forces and the effective normal force 
at the base of the slice. Therefore any assumptions made 
with regard to the side forces would enable determination of 
the effective normal force acting on the base of each slice. 
An assumption that simplifies the calculations the most 
was first proposed by Fellenius (1927). He assumed that the 
interslice forces acting on each slice have no net effect. 
That is, the effective normal and shear interslice forces 
acting on the left side of each slice are equal in magnitude 
but opposite in action to the respective interslice forces 
acting on the right side (Taylor, 1937). 

When no external side forces act upon the outside edge 
of the two end slices, the usual case for slope stability 
problems, the assumption requires that the interslice forces 



28 



acting upon these two slices be zero. Conformity to 
Newton's Third Law, stating that every action is always 
opposed by an equal reaction, requires that the remaining 
interslices also be zero, which is physically unreasonable. 

A water filled tension crack at the top of a slope is 
a situation where an external force would act upon the side 
of an end slice. For this case it is generally impossible 
to satisfy Newton's Third Law with the Fellenius assumption. 
Elimination of the side forces and the line of thrust 

from consideration, reduces the unknowns by a total of 3n-3. 

The number of unknowns are thus reduced to n+1, making the 

problem highly overdeterminate. 

The n force equilibrium equations obtained by summing 

forces normal to the base of each slice, while ignoring the 

side forces, are used to obtain the effective normal force 

acting on the base of each slice, 

AN' = AW cos a (14) 

Including the overall moment equilibrium equation, n+1 
equilibrium equations are used. 

The expression for the factor of safety becomes, 
n 
I (AC a + AW cos a tan <f>; 



F " " n" ~ (15) 



J AW sin a 

This expression does not satisfy force equilibrium parallel 
to the base of each individual slice. Also, since it is not 



29 



necessary to know or to assume a position for the effective 
normal force at the base of each slice to obtain a solution, 
satisfaction of moment equilibrium of each individual slice 

can not be shown. 

o 
The procedure as described above is known as the 

Ordinary Method of Slices. It is also called the Fellenius 
Method or Swedish Circle Method. The values of the factor 
of safety calculated by the above expression have been shown 
to be smaller than those calculated by the lower bound 
solution (Whitman and Moore, 1963). The error range of 
calculated values of the factor of safety may vary from five 
to forty percent from more accurate methods to be described 
later. The variation depends upon the value of the factor 
of safety, the subtended angle of the circular shear surface, 
and the magnitude and distribution of pore pressure when 
considered (Whitman and Bailey, 1967). it may be concluded 
then, that the assumption of no side forces used by the 
Ordinary Method of Slices is inaccurate. The unreasonably 
low values of the factor of safety could lead to conser- 
vative and uneconomical designs. 

Alternative expressions for the factor of safety can 
be obtained by use of overall horizontal or vertical force 
equilibrium. But it has been shown (Spencer, 1967; Wright, 
1969) that the value of the factor of safety by overall 
moment equilibrium is not as sensitive to the side force 
assumption. 



30 
Simplified Bishop Method 

Bishop (1955) proposed an alternative method for ob- 
taining a value of the factor of safety without including 
the effect of interslice forces. Overall moment equilibrium 
is satisfied as with the Ordinary Method of Slices. However 
the expression for the effective normal force at the base of 
each slice is arrived at in an alternate manner. 

Forces are summed vertically to obtain an equilibrium 
expression containing the effective normal force for each 
slice (Figure 5), ignoring the side forces, 

AW - AS r sin a - AN' cos a = (16) 
rather than summed normal to the base of each slice. By 
substituting the expression for required shear resistance 
(12) and solving, the resulting expression is obtained for 
the effective normal force at the base of each slice. 

AW - AC' sin a / F 
AN' = * _ (17J 

cos a (1 + tan $' tan a / F) 

Substitution of this expression into the overall 
moment equilibrium equation (11) yields an implicit ex- 
pression for the factor of safety. 

n 

I [(AC ^ cos a+AW tan <^) / cos a (1 + tan ^ tan a / F) ] 

"" n " (18) 

I AW sin a 

1 

The solution for a value of the factor of safety 
requires an iterative approach, assuming a value for the 



n 
F = 



31 

factor of safety and testing for equality. Although the 
Simplified Bishop Method, also called the Modified Bishop 
Method, does not provide a direct method for obtaining the 
factor of safety as does the Ordinary Method of Slices, the 
values obtained agree much closer to those obtained by more 
accurate methods considering interslice forces (Whitman and 
Bailey, 1967) . 

Bishop's Method, Rigorous 

A rigorous method capable of analysing circular shear 
surfaces for slopes composed of soil having both frictional 
and cohesive components of shear strength was first proposed 
by Bishop (1955). The method satisfies complete equilibrium 
and provides a means for judging the reasonableness of any 
solution arrived at. 

When arriving at an expression for the effective normal 
force acting on the base of each slice, the interslice 
forces are not ignored for this rigorous method. Forces are 
summed vertically to obtain the following equation of equili- 
brium for an individual slice (Figure 5) , 

AW + AX - AS r sin a - AN' cos a = (19) 

where AX is the difference of magnitude between the two 
shear forces acting on the vertical sides of the slice. 

Substituting the expression for shear resistance (12) 
into this vertical equilibrium equation and solving for the 
effective normal force, the following expression is obtained, 




32 



(20) 



Substitution of this expresgxon ^^ ^^ ^ ^ 

all moment equilibrium, equation (11) vieldq 

for fh. * Y 3n ex P re ssion 

for the factor of safety, 



n 



I A C ^ cos a + (AW + AX ) tan ., 
F = = r a 

~ (21) 




If internal equilibrium is to be 8ati8fied# ^ 

internal forces must satisfy the f nll • 

,. fy uhe fol lowmg boundary condx- 

tions . 



tions, 



n 

I AX = X - X 

1 o n (22a) 

n 

I AE' = e> - E' 

1 o n (22b) 



« - „■ is the difference ^ ^^^^ bet _ n the 

: and x ... Xn and E , are shear and * 

— es of the 3U ding .... respeotively (pigure " 
a, Slmming foroes paraUei ^ ^ ^^ ^ • 

« equation for force Milil , . 

of the .- rCee9Uillb "™ W directions exclusive 

of the vertical direction „ 

tl0n ' P rev «"=ly used, is obtained, 

<AW + iX) Ei « « - 4E' cos o - AS . 



33 




LJ 

y 

_j 

i 

UJ 

UJ 

I 

h- 

o 
o 

f- 

LJ 
(J 

£ 

§ 

ID 
O 
CD 



UJ 

r> 
u. 



34 



Solving for AE * , 

AE' = (AW + AX) tan a - AS sec a (23) 
and summing for all slices, 

n n 

I AE' = I [(AW + AX) tan a - AS r sec a] 

an expression satisfying equation (14b) is then arrived at, 

n 

I [(AW + AX) tan a - AS sec a] = E' - E' 124) 
1 r n o ; 



on 



Vertical force equilibrium, force equilibrium in a directi 
other than vertical for each slice, and overall moment 
equilibrium will be satisfied if appropriate values for AX 
can be selection to satisfy expressions (21), (22a) and (24). 
If not, complete equilibrium cannot be satisfied. 

Moment equilibrium of the individual slices was not 
considered when arriving at a solution for the factor of 
safety. Use of n-1 equations for individual slice moment 
equilibrium will provide the location of the line of thrust. 
Since overall moment equilibrium was considered, one equation 
for slice moment equilibrium is redundant and therefore not 
used. 

The position of the effective normal force, AN', at the 
base of each slice must be assumed in order to establish 
expression for moment equilibrium for each slice. Bishop 
(1955) assumed it to be at the center of the base of each 
slice. The weight of each slice, AW, was also taken at th 



an 



35 



center of each slice for moment equilibrium, as is implied ' 
by the overall moment equilibrium expression (3). 

If moments are summed about the midpoint of the base of 
each slice, the contribution of the weight, Aw, and the ef- 
fective normal force, An', to the resulting moment equili- 
brium equation is eliminated. The equation for moment 
equilibrium 

E' [(y- - y) - AX] 

" OB' + AE') [(y- + Ay .) - (y + Ay) + Av^ 

+ X $* + ( X + AX) Ax = (25) 

where A x is the width of the slice, and Ay and Ay^ ,re the 
incremental changes, across the width of the slice, of the 
functions defining the shear surface, y(x), and the line of 
effective thrust, y£(x), respectively. 

Solving for the location of the line of effective 
thrust at the right face of a slice, the following ex- 
pression is obtained, 

E' y? +AE' (y + Ail) + x . . y Ax 

(Yl + Ay*) = £ 2 ' A ax + AX — 

J t *t' ~ *- (26) 

E' + AE' 

The calculation of the location of the line of effective 
thrust is performed slice by slice in succession. 

For the first slice, the shear and effective normal 

side forces, X and E', are the boundary forces, x n and E' 

o 



36 



respectively. The location of the line of effective thrust 
at the left side of the first slice, y', coincides with the 
effective no™! boundary force E'. when % U abser .t, y- 
is at the base of the first slice's shear surface. 

The calculated values of x + 4X, E' + «■ , and v , + iy , 
become the values of x, E-, and y . for the next ^ ^_ C 
spectively. T he procedure is repeated until all slices are 
exhausted. The values of « for each slice are obtained 
from expression (23) . 

The position of the line of thrust provides an oppor- 
tunity to judge the reasonableness of a solution. There are 
an infinite number of choices for values of « which win 
satisfy complete equilibrium. „ the choice used produces 
a solution having a line of thrust totally within the slidin s 
»« implying no tens.cn wrthin the soil mass, the solution 
should be reasonable. 

Another criterion used to establish reasonableness of a 
solution satisfying complete equilibrium is whether or not 
excessive interslice shear stresses exist. The solution will 
not be acceptable if the interslice shear stress exceeds the 
strength of the soil at any vertical face. Satisfaction of 
this criterion is not generaHy sufficient. The most cri- 
tical internal shear stress generally occurs on an inclined 

plane (Seed and Sultan lQfi7^ t* ^ 

n ' 1967K If ^e shear strength 

-billed along a vertical plane is well below the available 
shear strength, an adequate solution can be assumed (whitman 
and Bailey, 1967) . 



37 



Bishop (1955) had found that althnn h 

at alt nough an infinite 

— « o, tt values uin satisfy cMpiete 

Thus verif ioation of B r afetY " »"9»ific«t. 

,. f reasonable l ine of thrust or satis 

factory interslice shear stress is * 
considering the effort „ ^ ^""^ 

satisfacf reqU " ed t0 achieve th -ou 9 h 

satisfaction of all criteria. 

When A x = o for an .i ■ 

SS ' ex P res sion ,13) reduces to 
the expression for «,«<=■•,. UMS to 

the Simpl ified BishQp Me 

Equation (14a, is satisfied but n 5) 

( 5) generally i s not 
Therefore complete e q uil lbrium ■ 

However the , ^^ 1S Orally not attained. 
However the value of the factor of safetv i, 
an , -arety i S reasonable 

and conservative compared with soluf 

— 3ht , Kulhawy , and ::: r zr :r e m - 

-- of the Simpllfied Blshop Method i • ' —"- 

*— - economic consecrations of desi gn . ^ 

Nonveiller MQfic;\ 

r (1365) extended Bishnn' c 
„,•, . aisnop s approach to con- 

-der shear surfaces of general shape The , . 

- Bishops when applied to .• , " "*""• 

PPHed to circular shear surfaces 

Overall moment equilibrium is estah, 
— *. about an arbitrary ooi t ^^ * 
is obtained " ' "" f ° 11 ~ 1 »» Session 



n n 



I AW x - J AS v V 

1 * f r x s ~ I AN' x n = o 



(27) 



where x , v -,-j 

„< * s . and are the 

resistance anrt « e wel S h t, shear 

' ^ eff ^tive normal force of ea ch sli 
spectiveiy ( Pigure „ _ EaCh 5ll oe re- 



38 




FIGURE 8 - \$E^fa*fflBLU* EXTENSA 



39 



Substituting equation (12), the expression for shear 
resistance, and solving for the faotor of safety the 
following expression is obtained, 



n 



I (A^ + AN' tar) <^> x< 

? H 

I AW x w - I AN - ^ 



s 
"" ~ ~ ~ (28) 



Substitution of expression (20, for the effective 
normal force derived by s» lr „ forces vertioally _ ^ 
Allowing expression for the factor of safety is obtained, 
? AC a cos o + (AW + ax) tan ♦' 
F =, 1 " >s a u + tan~*J-tiH~o~7-Fr *s 

f AW x - ? W ~ alr: ^" ^" a / F (29) 

1 " 1 L ' os o U + tan JT tan - ? Fj x n 

An acceptable solution must satisfy complete equili- 
brlU ™' MP " Ssi °- ("a) and (2.,, and also cr.terra pre- 
viously mentioned with regard to the line of effective 
thrust and interslice stress. 

Janbu's Generalized Procedure of slices 
'anbu , 19M . 1957 , and 19?3) deveiopea a ^^^ ^ 
analysing the stability of slopes assum.ng shear surfaces „, 
general shape. T he procedure, is based upon differential 
equations, which govern moment and force equilibrium of the 
*ass above an assumed shear surface. 

«o,e„t equilibrium is considered about ^ ^^ ^ 
the base of a slice of infinitesimal ^^ ^^ ^ 



40 




FIGURE 9 - , N F,N,TESIMAL SLICE AND FORCE SYSTEM. 



41 



The weight of the slice, dW, and the effective norma, force, 
dN«, acting on its base are both assumed to act through the 
midpoint of the base. Thus their contribution to the re- 
sulting moment equilibrium equation is eliminated. An 
equation in terms of differentials similar to equation (25), 
which was derived for a slice of finite thickness, is ob- 
tained, 

E'[(y; - y) - §*] - (e' + dE ') [(yj + dy ^) _ (y + dy) + ^ 
+ x §* + (x + dx) f2L „ o (30) 

where dx is the width of the slice, dE' and dX are changes 
across the slice of the values for the internal forces E' 
and X, respectively, and dy and dy^ are the changes across 
the slice of the functions defining the shear surface, y (x) , 
and the line of effective thrust, y-(x), respectively. 
Simplifying equation (30), it becomes, 

E ' ^r + ar- K + *y; - y - §*] 



dy' 

,, y t , dE' 

iy t 

dx 



+ dX H* - x = (31) 

As dx approaches zero, this equation approaches the 
following limit, 



dx~ + E dx~ - X = ° (32) 



The differential equation governing force equilibrium 
is obtained from equations satisfying vertical and horizontal 
force equilibrium of an infinitesimal slice (Figure 9), 



dW + dX - dN' cos a - dS sin a = 



r 



42 



(33a) 



dE* + dN' sin a - dS cos a = (335) 

where dS r is the resisting shear force at the base of the 
slice. 

Eliminating dN from both of these expressions, the 
following differential expression is arrived at, 



dS 
dE' 



cos a 



- (dW + dX) tan a (34) 



Considering vertical force equilibrium, an expression 
for the effective normal stress a^ acting on the base of an 
infinitesimal slice can be arrived at, 



, _ dW t dX 

n dx dx~ ~ s r tan a (35) 



where o^ = — and s r is the shear strength mobilized, 

dS c' + 0' tan 6' 

s = £ _ a n T a 

r dx f~ (36) 

Substituting the expression for the effective normal 
stress (35) into the expression for the shear strength re- 
quired to be mobilized for limiting equilibrium (36), and 
solving for the required shear strength, 

. = C a + ( dx~ + dx- } t an * a 

r T~77~TZ < 37 ) 

F (1 + tan a tan 0' / F) 

Substitution of this expression for the shear strength 
into expression (34) yields the differential equation 
governing force equilibrium, 



43 
c i . ,dW , dX> 

ds< = a ( gt + BT> tan ♦; dx 

2 ,dW dX. 

F (1 + tan a tan <f>' / F) cos 2 a dx + 3Ge tan a dx 

(38) 
Satisfaction of overall horizontal equilibrium and 
boundary conditions requires that, 

n 

/ dE = E' - E' , 

o o ^n (39) 

Therefore stability requires that, 
n c' - (2lL 4. "X. 

/ I- 2 ** £*- - a - c3x dw dx, 

o F (1 + tan ^ tan a / F , cos 2 a " ( dx" + dx"> tan a dx] 

= E ' - E ' 

o n (40) 

where E n is usually zero. 

Fro, this expression the factor of safety can be solved 
for, using finite difference approximation of the integrals, 

? c a Ax + < A W + AX) tan *' 

2 2 a 

F = 1 COS a (1 + tan a tan ^ / F ) 



? rA " "" " (41) 

2 [AW + AX] tan a + E' - E' 
1 o n 



The solution for the value of the factor of safety is 
as follows. An initial value of the factor of safety is 
obtained fro, expression (41, assuming that ta is equal tQ 

ly expressed, a trial and error or iterative approach is 



used 



By assuming a line of effective thrust and using the 
x-itll calculated value of the factor of safety, eguations 



44 



(32) and (38) can be used to determine approximate values 
for AX. Recalculation of the value of the factor of safety, 
using equation (41) and the approximate values of AX, should 
produce a more accurate result. The process is repeated un- 
til the value of the factor of safety converges to a 
reasonable degree of accuracy. 

The procedure will generally converge within a reason- 
able number of iterations. However, occasionally the 
solution will not converge. This could be a serious in- 
convenience. The only information available in this in- 
stance would be the initial value of the factor of safety 
calculated using the assumption that AX = 0. When the 
solution does converge, the procedure as outlined is more 
efficient than other methods considering side forces, be- 
cause the solution arrived at is based upon an assumed 
reasonable line of effective thrust. 

Morgenstern and Price Method 

Morgenstern and Price (1965) approached the problem of 
slope stability analysis in a somewhat different manner, 
developing a procedure capable of analysing shear surfaces 
of general shape. The approach also requires formulation of 
differential equations, governing equilibrium of the mass 
above an assumed shear surface. The differential equation 
governing moment equilibrium is identical to that derived 
earlier for Janbu's generalized procedure of slices method, 



45 

( *i - »> § + E ' anr - x - ° '32) 

Force equilibrium of individual slices is satisfied for 
directions normal and parallel to their bases (Figure 9) , 
rather than in vertical and horizontal directions as con- 
sidered by Janbu's generalized procedure of slices, 

dN' - dW cos a - dX cos a + dE ' sin a = (42a) 

dS r - dE' cos a - dX sin a - dW sin a = (42b) 

The resisting shear force, dS r , can be expressed in 
terms of the following Mohr-Coulomb strength relationship 
containing the definition of the factor of safety with re- 
spect to shear strength, 

dS r = F (c a dx / cos a + dN ' tan <j, ' ) (43) 

By substituting this expression into equation (42b) , 
solving for dN' , and substituting the resulting expression 
for dN' into equation (42a), the following differential 
equation is arrived at, 

c' 

—■ [1 + (Q-) 2 ] + tan *a r dW dX dE' dy, 

f lx w J + — f — [ dx" + dx- ~ set ai ] 

_ dF/_ , dXdy dW dy 

dx dx dx + dx" dx < 44 > 

where ^ = tan a. 

Equations (32) and (44) are the two governing differ- 
ential equations satisfying moment and force equilibrium for 
infinitesimal slices. They contain three unknown functions 



46 

E', X, and y^ for the internal effective normal and shear 
forces and the line of effective thrust, respectively, re- 
quiring additional assumptions. 

Morgenstern and Price (1965) assumed a relationship 
between the internal forces, 

X = X f (x) E' (45) 

where A is a scaling factor, and f(x) is a function of 
position, which defines the variational relationship of the 
internal shear force, X, with the effective normal internal 
force, E'. The function f is arbitrarily selected, but is 
acceptable only when the values of the solved unknowns are 
reasonable. 

To arrive at a solution, equations (32) and (44) are 
integrated, over the range of x defining the potential 
sliding mass, to test for satisfaction of boundary condi- 
tions. To simplify integration, the mass is divided into 
slices of finite thickness. Each slice is linearized; the 
base of each slice is linear; the ground surface at the top 
of each slice and interfaces between different soil types 
within each slice are linear, so that the weight vanes 
linearly across each slice; and the function f is assumed to 
vary linearly across each slice. The slice divisions are 
treated as positions of discontinuity. To integrate over 
the total mass, integration is performed over the individual 
slices, in turn. 



47 

Examining a typical linear slice, the following is ob- 
served. The shear surface can be expressed as a linear 
equation, 

y = A x + B (46) 

where A and B axe the slope and intercept of the slice base. 
The first derivative of the weight with respect to position 
within the slice can be expressed as, 

%£ ■ P x + q (47) 

where p and q are the slope and intercept of the linear 
relationship. The function f can be expressed as 

f = k x + m (48) 

where k and m are the slope and intercept of a linear re- 
lationship of the ratio of internal forces within a slice. 

Substituting equations (45), (46), (47), and (48) into 
equation (44) , the following equation can be arrived at, 

dF ' 
(K x + L) ~- +KE' =Nx + P (49) 



where 



K = - A k (tan <J» • / F - A) (49a) 

cl 

L = - A m (tan 4>'/F-A) + 1 + A tan <b • / F 

a a 

(49b) 
N = p (tan <f>' / F - A) (49c) 

c 

P = p^(l+A)+q (tan ♦ ' / F - A) (49d) 
If equation (49) is integrated over a linear slice, 



48 

then / 

2 
E' = [ E j_ L + ^L. + P X ] / ( L + N x) (50) 

where E| is the value of E' at the left edge of the slice, 
and x is a position along the slice measured from its left 
edge. 

Starting at the beginning of the shear surface, with 
E| equal to the known boundary value, E q , which is usually 
zero, the value of E' at the right edge of the first slice 
can be calculated with equation (50). This value, in turn, 
is used as the value of E' for the next slice. The inte- 
gration is continued in this manner, until all the slices 
have been exhausted. Then, if the boundary conditions are 
satisfied, E\ at the end of the shear surface will equal 
the known boundary value of E^, which is generally zero. 

If complete equilibrium is to be satisfied, boundary 
conditions with respect to internal moments must also be 
satisfied. Equation (32) can be integrated over each linear 
slice, providing an expression for the internal moment. 

M = E ' <Yt - y) " / <* - b> |£, dx (51) 

o 

The boundary conditions with respect to internal moments are 
satisfied when the integral over the entire sliding mass is 
equal to the known boundary value, H^, which is usually zero. 

If the boundary conditions are satisfied for equation 
(51), then the same equation can be used to determine values 
of y; to judge the reasonableness of the solution. 



49 



The solution requires assuming the function f and an 
initial set of values for F and A. Then integration is 
performed across all of the slices, slice by slice, to deter- 
mine E n and M n at the end of the shear surface. If the 
boundary conditions are not satisfied, then, by an iterative 
approach (Morgenstern and Price, 1967), new values of A and 
F are chosen until the boundary conditions are satisfied. 

When this is accomplished, reasonableness of the calcu- 
lated line of effective thrust and of the magnitude of 
interslice shear forces is examined. If either is judged 
unsatisfactory, the entire procedure must be repeated, be- 
ginning with the selection of a new function f, until a 
satisfactory solution is arrived at. 

Spencer's Method 

Spencer (1967) developed a procedure satisfying complete 
equilibrium, assuming a circular shear surface and parallel 
resultant interslice side forces. The method is easily ex- 
tended to assumed shear surfaces of general shape, and to 
other assumptions with regard to the slice side forces 
(Wright, 1969). The effective normal and shear side forces 
acting on both sides of each slice are replaced with statical- 
ly equivalent resultant side forces, Q, (Figure 10). Each is 
assumed to act concurrently, through the midpoint of the base 
of each respective slice, with the other slice forces; the 
weight, AW, the effective normal force, AN', and the re- 
sisting shear force, AS . 



50 




FIGURE 10 - SLICE AND RESULTANT OF THE SIDE FORCES. 



51 



Suction of forces, normal and tangent to ^ ^ ^ 
*ach slice, provide two e quatl ons of force e^ilibri™, 

iN ' " aw cos c + sin (a _ e) = o (52a) 

^S r -»sin.-o cos (a . e) . (52fa) 

where e is the initio of the resultant side force ^ 
-ch slice. Pro™ elation ,52a, an expression for the ef- 

obtained, 

AN ' = '" S1 » « * sin (c, - 6) (53) 

Substituting this expression into that for the re- 

substitutin g the express.on obtained into elation (52b) 
the resultant side forces, 0, can be solved for 




(54) 



K the overall no„t produced about an arbitrary point 
by the external forces is 2 ero, then the overall m o,ent of 
the internal forces m ust also be zero. Then , 



n 

I Q R cos ( a - 6 ) = 



(55) 

"here . is the distance fro m the point, about which _s 
are svu-ed, to the center of each slice. 

Also, if overall force equilibrium is to be satisfied, the 
station of internal forces in two mutually exclusive 



52 

directions must be zero r^ u 

zero. lor horizontal and vertical 

directions 

n 

I Q cos o = o ,_. . 

1 (56a) 

n 

I Q sin e = ,_,.. 

1 (56b) 

If the resultant side forces ^ „ 

rorces are assumed to be parallel; 
e.g., 6. = constant, equations tst:\ u 

i quations (56) become identical and 

can be expressed as, 

n 

Iq = o 

1 (57) 

The inclination of the resulf.nf ■* * 

resultant side forces could 

also be expressed as 

e i ■ e f(x) (58) 

wW o isa scaling angle of inciination ^ f w ^ ^ 
- ributlon deflnlng how ^ ^^^ ^^ positiQn x (wr ^ h ^ 

"69,. Parallel side fQrces ^^ ^ ^^^ ^ ^ 
range of x. 

The p aralle l resultant side force as _ ption ^ ^^^^ 
ent to the Morg ensta rn a nd Price Hethod ,„„, ^ ^ = 
• Then spencers t a „ e is egUivalent tQ Horgenstern , s and 
Price's A (Spencer, 1973) . 

UniqUS ValU6S *» the '«*« °t «f.*y. P, and the 

;T7 inCllnati ° n ° f the — * ■«. «««.. e. can be 

■<*v* for , if eguations (SS) ^ (56) ^ sati ^ ie ^ 



53 



equations (55) and (57) apply if f ( x) = 1 for the range of 



x. 



Reasonableness of the solution can be judged by the 
position of the line of thrust and the magnitude of the 
interslice shear stresses. Both are obtained from the 
moment equilibrium equations for the individual slices (25) 

Spencer's Method requires less computation time than 
the Morgenstern and Price Method and accomplishes the same 
task of attaining complete equilibrium, using an assumed 
relationship between the internal forces. Both require 
attention to solved unknowns to check for their reasonable- 



ness . 



Carter's Method 

Carter (1971) demonstrated that if Ax is assumed to be 
zero for each slice, the value of the factor of safety, 
determined by satisfying overall moment equilibrium and 
vertical force equilibrium of the individual slices, varies 
with the position of the point about which overall moment 
equilibrium is satisfied. Nonveiller (1965) had not indi- 
cated this behavior, when he extended Bishop's rigorous 
method to surfaces of general shape and suggested simpli- 
fying the calculations by assuming AX = 0. Overall moment 
equilibrium had been satisfied about an arbitrary point. 

By summing moments about an arbitrary point (Figure 
11), the following expression for moment equilibrium is 
obtained. 



54 




FIGURE II- 



?ko£J: 0RCES REFERENCED TO AN 
ARBITRARY POINT. rtlN 



55 

n 

J [AN 1 (y S i n _ ~ cos oJ + w - 

- AS r ( x sin a + y~ cos a)] = (59) 
"here x and y are the horizontal and vertical distances 
from point to the midpoint of the base of eaoh slice. 

Expression (12) for the resisting shear force, ,3 is 
substituted into equation (59). Then by substitute ' 
expression (20) for the effective noraal force, »».. as 
determined by satisfying vertical £orce equilibrium ^ 
each siice, rearranging l ike terms, simplifying and trans . 

Posing the A X term, an expression ™„* • • 

expression containing n+1 unknowns, 

F and n values for A X, is arrived at. 

n _ AC; / cos a + AW (tan 0' - F tan ) 

I Y - 

F + tan a tan <*, ' 

T a 

n F tan a - tan <j> ' x 

I y ax [ ____________ _ 

1 F + tan tan ^ J (60) 

If the assumption, AX - 0, is made, the expression 
reduces to, 

n _ AC; / cos a + W (tan *• - p tan a) 

I y ,_ n 

1 F + tan a tan «, ■ * ° (61 > 

^a 

The factor of safety is implicitly expressed. Although its 
value is independent of the horizontal position of the pent 
about which moments are summed to achieve overall m oment 



56 



equilibrium, it is dependent upon the vertical posit-on 
of that point. 

Carter (1971) found that as y goes to infinity, 
the value of the factor of safety approaches a minimum 
asymtotically. At infinity, y can be taken out of the 
summation, 

n AC; / cos g + AW (tan ^ - f tan a) 



1 



F + tan a tan <jP " =0 (62) 



a 



This expression satisfies vertical equilibrium of the 
individual slices and overall horizontal force equilibrium. 
It does not satisfy moment equilibrium of the individual 
slices. The value of the factor of safety is conservative 
with respect to reasonable solutions obtained by more 
accurate methods satisfying complete equilibrium. The 
method enables evaluation of general shear surfaces with a 
minimum amount of computation time. 



57 



METHOD CHOSEN FOR COMPUTER PROGRAM 

Of the limiting equilibrium models discussed, those 
which divide the mass above the assumed shear surface into 
slices are most ideally suited to being programmed to solve 
general slope stability problems. The Ordinary Method of 
Slices is restricted to circular-shaped shear surfaces, 
and may provide a very conservative evaluation of the 
stability of a slope. For these two reasons, it was not 
considered appropriate for the program developed. The 
Simplified Bishop method, although of sat 1S factory accuracy, 
is also limited to circular shear surfaces, and therefore 
was not considered. 

Although the Rigorous Bishop method (1955) also evalu- 
ated only circular shear surfaces, the extension of Bishop's 
work to consideration of general shear surfaces by 
Nonveiller (1965) gives the ty F e of model that is desired. 
Difficulty is encountered, however, when selecting values 
of AX for each slice to satisfy the conditions required for 
limiting equilibrium. After selecting a satisfactory set of 
values for AX, a reasonable line of effective thrust and 
reasonable interslice shear stresses must be verifred. If 
not, a new set of AX values, satisfying limiting equilibrium 
conditions, must be selected, and the cycle is repeated. 



58 



The process of obtaining a satisfactory solution can become 
lengthy , since there is no systematic procedure of 
selecting better values of 4X for each additional cycle 
of calculations. 

If ax is assumed to equal zero for each slice, as 
Nonveiller (1965, suggested as a simplified solution, 
complete equilibrium is not, in general, satisfied. The 
value of the factor of safety win vary with the ^^ 
of the point about which overall moment equilibrium is 
taken. It is known that for homogeneous problems evaluated 
assuming circular shear surfaces, satisfactory solutions for 
the value of the factor of safety are obtained if the over- 
all moment equilibrium is taken about the center of the 
circular surface. However for irregularly shaped shear 
surfaces, there is insufficient experience to draw such a 
conclusion. 

Janbu's generalized procedure of slices satisfies 
complete equilibrium, and is based upon an assumed reason- 
able line of effective thrust. The only criterion which re- 
quires checking is that for excessive interslice shear 
stresses. lf the interslice shear stresses are not satis- 
factory, a different line of effective thrust must be as- 
sumed, m addition, the solution may involve difficulties 
in convergence upon a unique value for the factor of safety. 

The Morgenstern and Price method (1965) offers no 
advantage over the use of the Spencer method as Wright ,1969, 



59 



has modified it. The Spencer method is sxmpler and revues 
signxficantly less calculation. The Mor g enstern and Price 
method requxres nun.er.cal integration over each slxce, which 
is a lengthy procedure. Both methods have the same advan- 
tage over that presented by Nonveiller (1965) , because 
numerical procedures can be used to arrive at an acceptable 
s.de force condition, assumxng a varxational relationshxp 
of postion between the interslice shear forces, f( x ). 
Reasonableness of the line of effective thrust and the 
magnitude of the xnterslxce shear stresses stxll require 
checking, and xf not satisfactory, a dxfferent variatxonal 
relationship of position between the interslice forces, 
f(x),must be assumed, until an acceptable solution is ob- 
taxned. This can sometimes require substantial effort. 

Carter's method (1971), capable of consxderxng assumed 
shear surfaces of general shape, does not satisfy complete 
equxlxbrium, but does provxde values of factors of safety, 
which are reasonably accurate. The result xs always conser- 
vatxve. The amount of calculatxon xs mxnxmal, and is com- 
parable to that of the Simplxfxed Bishop method. There is 
no need to check the reasonableness of the solution, as the 
solution is conservative with respect to more accurate 
methods. For use in routine slope stability analysis, the 
method is practical and has therefore been incorporated 
within the program developed. 

Carter's method, derxved earlxer for fundamental 
forces, is expanded here to consxder additional forces, 



60 



Which are fluently encountered in slope stability probes 
The forces acting on a typical slice are shown in Figure 12 
The geometry of each slice is described by its height, h 
measured along its centerline. its width, Ax, and by the 
inclination of its base and top, „ and 8, respectively. 

Overall foment equilibria, about an arbitrary point. 0, 
is expressed as, 

n 

I [(AN' + AU a ) (y- sin a - x cos a) 

+ ^U 6 (x cos - a sin 6) 

+ AQ (x cos 6 - i" sin 6) 

+ AW (1 - k ) x + fc W (y - h , 

~ AS r (x sin a + y cos a) ] = o (63) 

where ^ and 4Ug are the resultant forces of pressures 
produced by water at the base and top of each slice, re- 
spectively; ,o i. the resultant foroe of a Miformiy dist ^_ 

buted surcharge boundary load acting upon the top of each 
slice with « as its angle of inclination; k y and ^ are 
vertical and horizontal earthquake coefficients which re- 
late the magnitude of the earthquake force to the weight of 
each slice; „ eg is the distance from the base of each slice 
to the horizontal component of the earthquake force; and 
a = y-h. 

Vertical force equilibrium for each slice is expressed 



as, 



(AN 1 + AU ) cos n + ac 

uv a / cos a + AS r sm a + AX - AW (l - k ) 

V 

- AU cos 6 - AQ cos 6 = (64) 



61 



+ 





K AW f K v A W 



x+Ax 




E*AE 



X 



Au, 



FIGURE 12- FORCES CONSIDERED FOR COMPUTER 



62 



Expression (12) for +h~ 

r ** resis ting shear force, AS , is 
substituted into equations ffi?i „ , r 

q nS (63) and <64). By solving 
for the effective normal force an- • 

' AN ' ln equation (64), and 

3U bStltUting the resulting expression int<> e ^ ation 

*- ^ e 9uation oontaining n+i unknowns _ p ^^ 

AX values, is obtained. 

" A. - F a, n 4. a „ 

1 F + A, = 2 y AX [— - ^a / * x 

3 1 l + tan a tan <p • / f " r- 1 (65) 

where, 

AC a 

A l = + tan • (aw r-ku q _ h e, , 

cos a a h VJ - z~ ) tan a + (i _ j, \ 

y v 

AU - 

" co^" + AU B (C ° S 2 + Z tan a sin &) 

y 

+ AQ (cos M ! tan a sin 6)} 

y 

A 2=A» Id- V tan ♦ ^ (1 . * , 

y 

+ AU (tan ex cos e - ^ sin e , 



y 

a 



+ AQ (tan a cos 6 - - s in 6) 

y 

A 3 = tan a tan <p ' 

By using the Bishop simpl ifying assumptio ^ Ax = 
expression (65) becomes, 




(66; 



63 

As the y coordinate of the point, about which overall 
moment equilibrium is satisfied, approaches infinity, the 
relative differences between the y of each slice become 
insignificant and y can be considered identical for each 
slice. Considering this limiting condition, y~ is taken 
out of the summation so equation (66) becomes, 

n A 1 - F A 2 

I F + A 3 = ° (67) 

Also as y approaches infinity, the values of h AF and 

eq' J 

a/y approach zero and unity, respectively. Then the terms 
A^ and A 2 become, 

A l = C a 7 cos a + tan *i IW (1 - k v - k h tan a) 
~ u a / cos a + U g (cos 6 + tan a sin 6) 
+ Q (cos 6 + tan a sin 6) ] 

A 2 = W [1 - k ) tan a + k. ] 

+ Ug (tan a cos 6 - sin 0) 

+ Q (tan a cos 6 - sin 6) 

Equation (67) forms the working model programmed for 
the computer program developed herein. 



64 



PROGRAM STABL 



The actual calculation of the value of the factor of 
safety, although time consuming, comprises a relatively 
small portion of the program developed. Most of the pro- 
gramming effort was directed at handling the geometry of a 
general slope stability problem. a description of the 
program's capabilities, and instruction with regard to its 
use, is reported elsewhere (siegel, 1975). a discussion 
of how the program performs its functions is presented 
below. A listing of the program is provided in the Appendix 
A of this volume. 

£»i£ion of Mass into slices „h„„e_the Assumed shear Surf,., 

The mass above the trial shear surface is divided into 
slices with vertical interfaces. The interfaces occur at 
all specified coordinate points def ining georaetry Ulthin ^ 
sliding mass or the assumed shear surface (Figure 13). If a 
water surface has been specifred, the sliding mass is 
further divided at intersections of the water surface with 
the trial shear surface, and also at intersections of the 
water surface with subsurface boundaries within the sliding 
mass. Slice divisions are also located at the intersections 
of subsurface boundaries with the tria! shear surface. 



65 




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Further divisions of the sliding mass into slices is 
not required, because doing so does not increase the ac- 
curacy by Carter's method of slices. 

After the sliding mass has been divided into slices as 
outlined previously, the following characteristics of each 
slice can be noted. The top and bottom are single straight 
line segments. If more than one soil type exists within a 
slice, the interfaces are also single straight line segments. 
And if a water surface transverses a slice, the surface is 
also a single straight line segment. If a boundary sur- 
charge load exists at the top of a slice, the load extends 
the full width of the slice, and its magnitude and direction 
is constant. The sliding mass has been sliced such that 
each slice is linear with respect to its geometry and 
boundary loading. 

Total Weight Calculation for Each Slice 

Recognizing that each slice is a trapezoid with vertical 
parallel sides, the calculation of the total weight for each 
slice is simple. The end slices constitute a special case 
of a trapezoid, when one of the vertical parallel sides has 
no length (Figure 14), i.e., a triangle. 

If a slice is subdivided by a subsurface boundary, the 
division is linear and extends across the entire width of 
the slice establishing subsections of the slice, which are 
also trapezoids with parallel vertical sides (Figure 15). 



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The total weight is calculated separately for each 
subsection, if the slice is subdivided. If not, the total 
weight for the slice is calculated as if the slice consisted 
of a single subdivision equivalent to the entire slice. 
Each subsection contains one soil type. If a subsection is 
totally above or totally below the water surface, the total 
weight of a subsection is equal to the product of the area 
of the subsection and the respective moist or saturated unit 
weight. Units are consistent, since the weight is in terms 
of a unit thickness normal to the plane containing the 
problem's profile. By multiplying the difference in ele- 
vation of the top and bottom boundaries of the subsection, 
measured at the slice centerline, by the width of the 
slice, Ax, the area of a subsection can be calculated. The 
elevation difference measured at the slice centerline is the 
average of the length of the two parallel sides of a 
trapezoid. 

If a subsection is partially submerged, the subsection 
is divided into submerged' and drained portions. As a result 
of the method by which the sliding mass was divided into 
slices, the water surface will only intersect a subsurface 
boundary at a face of the slice (Figure 15(a)). However, 
the water surface could coincide with a subsurface boundary, 
making the subsection above entirely drained, and the one 
below entirely saturated (Figure 15(b)). Since the water 
surface will only intersect a subsurface boundary at the 



70 



face of the slice, both the saturated and unsaturated 
portions of a partially submerged subsection are trapezoids 
The total weight of each portion will be the product of the 
areas and their respective unit weights. The areas are 
calculated as described before. 

After the weights of the individual subsections are 
calculated, they are summed to obtain the total weight of 
the slice. 

Water Forces 

For problems when the slope is totally or partially 
submerged, each individual slice, which is totally sub- 
merged, has a force, AU , acting upon the soil-water inter- 

p 

face (Figure 16). The average hydrostatic pressure, acting 
upon the top of each submerged slice, is equal to the 
product of the unit weight of water and the average depth 
of water above the slice, which is measured at the slice 
centerline. The magnitude of the force AU is the product 
of the average hydrostatic pressure and the true length of 
the top of the slice. The force acts normal to the soil- 
water interface. Its position is not required for the 
factor of safety calculations. 

The pore pressure acting on the base of each slice is 
defined in a number of ways. First, if a saturation line 
has been specified, the pore pressure at any point beneath 
it may be related to it. The pore pressure at a point 
beneath the saturation line may be approximated by assuming 






71 




FIGURE 16- WATER FORCES ACTING UPON SUBMERGED 
SLICES. 



72 

that it is equal to the product of the unit weight of water 
and the vertical distance measured from that point to the 
line of saturation. 

In general, the procedure is conservative and reason- 
ably accurate (Figure 17). However, accuracy decreases as 
the slope of the saturation line increases. The result can 
be unconservative in the region beneath a surface of seepage 

Second, the pore pressure may also be defined as a 
ratio of the overburden pressure, r u a Q , where r is a homo- 
geneous pore pressure parameter (Bishop and Morgenstern, 
1960) and o q is the vertical overburden pressure. Finally, 
the pore pressure may also be defined as a constant value 
throughout a zone. A combination of any two or all three 
of these means of defining the pore pressure may also be 
used. 

The water force, AU , acting at the base of a slice, 
is then the product of the average pore pressure, determined 
at the midpoint of the base, and the true length of the base. 
Its direction is normal to the base of the slice, and again, 
its position is not required for the factor of safety calcu- 
lation. 

If a pseudo-static earthquake load is applied to a 
slice, the pore pressure at the base of the slice will de- 
crease or increase, depending upon the direction of the ap- 
plied force. The magnitude of the water force, AU , is 

a 

decreased by an amount equal to the component of the earth- 
quake force normal to the base of the slice, if the 



73 




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earthquake load is directed away from the base. If the 

earthquake load is toward the ba3e of the slice, AU is 

a 

increased. The force AU is allowed to be negative, but 
a minimum value, corresponding to the product of the 
cavitation pressure and the true length of the base, is 
maintained. 



75 



SEARCHING TECHNIQUES 

The number of slopes having subsurface profiles for 
which an engineer can intuitively select the most critical 
shear surfaces for analysis are few. For cases of slope 
profiles containing a layer or seam of weak strength, it 
may be obvious that the critical surface will pass through 
the layer. However, the position and orientation of the 
critical shear surface within a weak layer may not be obvi- 
ous unless the layer is thin. Even then, it may be neither 
obvious where the critical shear surface enters and exits 
the layer, nor what path it might take beyond these points. 
Quite a few trial shear surfaces may have to be analysed 
before an engineer can be reasonably assured of locating 
the critical surface. 

Charts have been prepared for locating the critical 
shear surface of circular shape for slopes having horizontal 
toes and crests and composed of homogeneous and isotropic 
material (Taylor, 1937; Morgenstern and Price, 1965; and 
Spencer, 1967). Their usefulness is confined to problems, 
which can be idealized, as such, without gross assumptions. 
To reduce the amount of time required to prepare data 
defining a large number of trial shear surfaces, three 
trial shear surface generators have been developed for use 



76 

with program STABL. The first, named CIRCLE, generates 
surfaces of circular shape. This generator is good for 
slopes comprised of roughly homogeneous and isotropic 
material. The second, named BLOCK generates trial failure 
surfaces of a sliding block character, and is most useful 
when a well defined weak layer exists. Finally, RANDOM 
generates surfaces of general irregular shape. It is most 
useful for hetrogeneous soil profiles where both the 
location and shape of the critical surface are uncertain. 
Each generator uses pseudo-random techniques for 
generating the surfaces. 

Random Numbers 

Before explaining how each trial failure surface 
generator works, the means of obtaining random numbers, 
which are required by each generator, is discussed. 

Pseudo-random numbers are generated in an exact 
deterministic process by which a number is generated from 
its predecessors in a series. The random number generator 
used for program STABL must be able to generate real numbers 
having magnitudes within the range between and 1. To be 
random, the numbers generated must be uniformly distributed 
within this range. A number generated by such a process is 
not truly random, but for most practical purposes it may be 
assumed so. 

Most computer systems have a library function for 
generating random numbers. Control Data Corporation has 



77 

such a function, named RANF(X), in its 6000 series system 
library. This function is referenced within program STABL. 
Relatively simple random number generators can be written 
for any system if not readily available (Ralson and Wilf, 
1962) . 

Circular Surface Generation 

A circular-shaped surface, composed of a series of 
straight line segments of equal length, is generated from 
left to right. The length of the line segments is speci- 
fied. The surface initiates from a point on the ground 
surface. 

To begin, a line segment is projected from the initi- 
ation point in a direction chosen between two limits. The 
direction limits are either specified or automatically 
calculated by the program. In the latter case the clockwise 
direction limit, « 2 , is -45° with respect to the horizontal, 
and the counterclockwise direction limit, is 5° less 
than the inclination of the ground surface to the right of 
the initiation point (Figure 18). 

The selection of the inclination, e, at which the 
initial line segment of a surface projects, is determined 
randomly. A random number, R, having a value between and 
1, is generated and squared. The squared random number and 
the difference in the inclination of the two direction 
limits, a± - a 2 ,are multiplied. The result is added to the 



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inclination of the clockwise direction limit, a , to 

obtain the angle of inclination of the initial line segment, 

6 = a 2 + (a x - a 2 ) R 2 (68) 

Squaring the value of the random number generated, be- 
fore multiplying it with the absolute value of the differ- 
ence of inclination of the two direction limits, a - a 

'1 2 ' 
introduces a bias in the random direction selection process. 

Although selection of any inclination of the initial line 
segment within the two limits is possible, it is more likely 
that the initial line segment be located nearer the clock- 
wise limit. 

Figure 19 indicates the probability that an initial 
line segment will be oriented within any of ten equal ranges 
of direction, which subdivide the complete range of orien- 
tations. If the process of selecting an initial line 
segment were repeated one hundred times, it could be ex- 
pected that for approximately 32 times the initial line 
segment would be oriented within the subdivision next to 
the clockwise limit, while only 5 times would it be within 
the subdivision next to the counterclockwise limit. 

The primary reason for introducing the bias is to ob- 
tain a good distribution of completed surfaces. When the 
initial line segment was given an equal likelihood of being 
oriented in any direction within the direction limits, poor 
distributions of completed surfaces were obtained. 



80 



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After establishing the initial line segment of a trial 
failure surface, the remaining portion of the circular- 
shaped surface is randomly located, within some constraints. 
Although the exact location where the critical shear surface 
exits the top of the slope is not known, a zone where it 
probably will exit, can be assumed. Termination limits are 
established at the ground surface, so that only surfaces 
of interest are considered. 

The initial line segment is assumed as the chord of a 
family of circles (Figure 20). Only a subset of this 
family of circles is of interest. The circles having 
centers below the initial line segment are not considered 
for obvious reasons. Of the remaining circular surfaces, 
only those which lie within the zone bounded by the circu- 
lar surfaces passing through the two termination limits 
are of interest. 

For a circular-shaped surface composed of straight 
line segments of equal length, T, each successive line 
segment is deflected by a constant angle, A6, with respect 
to the line segment preceding it. The two limiting de- 
flection angles, AB^ and 48^ (Figure 20), which bound 
all the circular-shaped surfaces of interest are back- 
calculated (Figure 21) , 

A0 = 2 sin" 1 (T/2r) (69) 

where r is the radius of the circle. The same procedure is 
repeated for the other termination limit. 



82 




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Using the two deflection angles back-calculated as the 
deflection limits, a deflection angle, A6\ is randomly 
chosen from within the range defined in a fashion similar 
to selection of the direction of the initial line segment, 

AG' = (AG - A6 . ) R 2 (in) 

max mm' (/U) 

The orientation, e, of each succeeding line segment is equal 
to the inclination of the preceding line segment, plus the 
randomly selected deflection angle, 

e. = e.^ + AG' ■ (71) 

There are a number of circumstances when the procedure 
generates unsatisfactory circular-shaped surfaces, and ad- 
justments to the surface generation parameters, a-,, a , 
A6 max' and A6 min' become necessary. 

The minimum value of A6 min allowed is zero, i.e., a 
planar shear surface. If A9 nin is less than zero, it is 
adjusted to equal zero. Circular-shaped shear surfaces 
which are concave downward (Figure 22(a)) are not considered 
desirable. If the value of A9 max is also less than zero, 
all surfaces would be concave downward, and the inclination 
of the initial line segment must then be changed. If this 
is the case, the clockwise direction limit, a± , for select! 
of the new initial line segment is adjusted to the in- 
clination of the current initial line segment. A new 
inclination for the initial line segment is randomly 
selected between ^ and the new 0l using equation (68). 



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-A0(Not Permitted) 




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FIGURE 22- LIMITATIONS IMPOSED ON DEFLECTION 



86 

If the new inclination of the initial line segment is also 

unsatisfactory, A6 max is less than zero, the process of 

modifying a;L and selecting a new inclination for the initial 

line segment is repeated, until a positive value for A6 

max 
is detected. 

The maximum value of A9 mav permitted is arbitrarily 
set at 45°, the rationale being that a surface composed of 
line segments deflected at larger angles would be too crude 
an approximation of what is intended to be a smooth shear 
surface (Figure 22(b)). if A6 max is greater than 45°, it 

is adjusted to equal this value. If the value of A8 is 

min 

also greater than this value, the inclination of the initial 
line segment must be changed. In this case, the counter- 
clockwise direction limit, c* 2 , for selecting the new initial 
line segment is adjusted to the inclination of the current 
initial line segment, similar to the process seen before. 

Another adjustment which is sometimes required occurs 
when the shear surface overturns (Figure 23(a)). If for an 
initial line segment, the center of the circle passing 
through the left termination limit is below an elevation 
T/2 above the ground surface at that termination limit, 
A6 max is ad 3 usted to the deflection angle corresponding to 
a circle having a center at that elevation (Figure 23(b)). 

If the center of a circle passing through the right 
termination limit is also below an elevation T/2 above the 
ground surface at that termination limit, no satisfactory 



87 




(a) 




Perpendicular Bisector of Initial 
Line Segment 



(b) 



FIGURE 23- LIMITATIONS IMPOSED ON INCLINATION 
OF FINAL LINE SEGMENT. 



88 



surface can be generated with the initial line segment, 
inclined as it is. A new inclination is selected randomly 
for it, using equation (68), after adjusting the clockwise 
direction limit, a 2 , to the present inclination of the 
initial line segment. This may have to be repeated until 
a satisfactory inclination for the initial line, segment is 
selected. 

If a limitation is imposed with regard to a maximum 

depth at which a surface is desired, the value of A6 . can 

min 

be adjusted to accommodate this limitation, if necessary. 
If the minimum angle of deflection meeting this limitation 
is greater than AS^, then a new inclination is randomly 
selected, modifying a 2 first as before. 

One final form of adjustment of the surface generation 
parameters for circular-shaped surfaces is encountered with 
respect to surface generation boundaries which can be es- 
tablished. Line segment boundaries are established with the 
intention that no shear surface should pass through them. 
Each individual boundary is designated such that if inter- 
sected, an attempt is made to complete a surface using the 
same inclination of the initial line segment, either above 
or below the boundary. 

If an intersected boundary is designated as an upward 
deflecting boundary, a surface having a deflection angle 
equal to A0 max is checked for possible intersection with the 
same boundary. If it does intersect, a new inclination of 



89 

the initial line segment is selected without modification to 
the direction angle limits a ± or a^ If it does not inter- 
sect, A6 min is changed to the deflection angle of the 
surface which originally intersected the boundary, and a new 
deflection angle is randomly selected. The proceding is 
repeated if necessary until no intersection with any 
boundary occurs. 

General Irregular Shear Surface Generation 

The inclination of the initial line segment for a 
general irregular shear surface is selected in the same 
manner as described for the circular-shaped shear surface. 
However, the direction of each subsequent line segment is 
determined independently of the line segments preceding it. 

The counterclockwise limit for the direction of a line 
segment, A9 1 , is normally deflected 45° counterclockwise 
from the projection of the preceding line segment (Figure 
24). if for a particular line segment the orientation of 
the counterclockwise direction limit is greater than 90° 
(measured with respect to the horizontal), the inclination 
of the counterclockwise direction limit, 6^ is adjusted to 



90 



The clockwise deflection limit for the direction of a 
line segment, A6,,, is randomly selected for each shear 
surface generated, 



A6 2 = R 2 45° (72) 



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If for a particular line segment the inclination of the 
clockwise direction limit, 9^ is less than -45% it is 
set at this value. 

The inclination of a line segment is then established 
by, 

• - * a + M 1 - 8 2 ) R <1 + *) (73) 

where the angular extent of permissible directions, ^ - ,_,, 
is multiplied by a random number raised to a random power ^ 
between 1 and 2 and added to the inclination of the clock- 
wise direction limit, 6-. 

The above procedure, although somewhat arbitrary, does 
produce reasonable irregular surfaces of random shape and 
position. Reverse curvature of a shear surface is possible, 
but the frequency of occurrence is small. Unless very short 
line segments are used, "kinkyness" of the resulting gener- 
ated shear surfaces is not a problem. 

The other restrictions xmposed upon the circular-shaped 
surfaces are also imposed for irregular-shaped surfaces, 
but in a somewhat different manner. if a surface should 
intersect the ground surface short of the left termination 
limit, it is rejected, and a replacement surface is 
generated. 

If a surface should intersect the ground surface be- 
yond the right termination limit, or if an uncompleted 
surface extends beyond this limit, the possibility of success- 
fully completing the surface without rejecting the entire 



92 



surface is checked. First, the violating line segment is 
rotated to the inclination of the counterclockwise direction 
limit, e r which was used to select it. if at this in- 
clination the surface intersects the ground surface beyond 
the right termination limit, or it extends uncompleted be- 
yond this limit, the surface is rejected, and a replacement 
is generated. 

If a line segment extends below the maximum depth 
permitted, it is rotated to its counterclockwise direction 
limit. If it still extends below the depth limit, the 
entire surface is rejected, and a replacement is generated. 
If not, 6 2 is adjusted to the old inclination of the line 
segment, and a new inclination is selected as before. The 
procedure is repeated as necessary. 

When a surface generation boundary is intersected, the 
possibility of rotating the intersecting line segment clock- 
wise or counterclockwise (the direction depending upon the 
designation of the boundary) to clear the boundary is checked. 
If it can clear, a new direction is selected for the line 
segment in a fashion similar to that cited before. 

Searching for the Criti cal Circular or General Shear Surfaces 

Trial shear surfaces are generated from a number of 
initiation points with equal horizontal spacing along the 
ground surface. For example, Figure 25 shows ten circular- 
shaped trial shear surfaces generated from each of ten 
initiation points; a total of one hundred surfaces. As the 



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surfaces are generated, their respective factors of safety 
are evaluated, and the ten most critical are identified 
(Figure 26) . 

The ten most critical surfaces will in most cases fo 
a distinctive critical zone. Occasionally, more than one 
critical zone can be distinguished. In this case each zone 
could be considered individually. 

The compactness of the critical zone, its location 
within the zone containing all the surfaces generated, and 
the magnitude of the range of values for the factors of 
safety of the ten most critical surfaces, indicate the like- 
lihood that a shear surface exists with a factor of safety 
significantly lower than any calculated. If the critical 
zone is narrow, if non-critical surfaces have been generated 
on both sides of the critical zone, and if the values of the 
ten most critical surfaces are nearly the same, the defined 
factor of safety of the most critical surface generated 
could be assumed, with reasonable confidence. On the other 
hand, if the critical zone is wide, and/or the more critical 
surfaces lie along one edge of the zone containing all the 
surfaces generated, and/or the magnitude of the range of 
factor of safety values for the critical surfaces is large, 
the possibility of a shear surface with a significantly 
smaller value for the factor of safety may require generation 
of additional surfaces. 



95 



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If more trial shear surfaces are required, the limi- 
tations can be revised to restrict the additional surfaces 
to a particular zone of interest. For example, additional 
shear surfaces generated for the problem shown in Figure 26 
should all initiate from the trough of the bench on the 
slope. They should also be required to terminate at least 
15 ft behind the crest. 

In this particular probJem, the lower subsurface inter- 
face is defined by upward deflecting surface generation 
boundaries representing a competent slightly folded bedrock 
surface. Figure 25 demonstrates the control these boundaries 
have on surface generation. If additional surfaces were to 
be generated, a supplemental downward deflecting surface 
generation boundary could be defined. It would extend from 
the ground surface, somewhere on the face of the slope, to 
a point above the critical zone sufficient to prevent 
generation of trial shear surfaces through noncritical zones. 

Sliding Block Surface Generatio n 

If a zone of weakness within the mass of a slope is 
obvious, or if a critical zone has been fairly well defined 
by previous usage of the circular or irregular surface 
generators, it is often advantageous to use a third generator. 

A series of boxes, a minimum of two, i s located along 
the path of the zone to be investigated. The boxes are 
parallelograms with vertical sides. They can be specified 
to bracket the zone to be investigated (Figure 27). 



97 




I " 

i /^Extent of 
Search 



Intensive Search of Critical Zone Previously 
Defined by CIRCLE or RANDOM 




Weak Layer 



Search in Irregular Weak Layer 



nGUREar.SL.DJNGB^CK GENERATOR US.NG MORE 



THAN TWO BOXES. 



98 
A point is randomly selected from within each box. Any 
point within a particular box has an equal likelihood of 
being selected. The points are then connected in sequence 
with straight line segments (Figure 27) . 

If two boxes are specified within the weak layer of 
the problem shown in Figure 28, and a point is randomly 
selected from each, the resulting line segment forming the 
base of the central block has random length, orientation, 
and position. 

Next, the active and passive portions of the trial 
shear surface are generated (Figure 29) . The active and 
passive portions are composed of Ixne segments of equal 
specified length. The inclination of each line segment 
for the active portion of the shear surface is selected in 
a biased random fashion between direction limits inclined 
at 45° and 90°, 

6 - 45° + 45° R 2 (?4) 

For the passive portion of the shear surface, the 
inclination of each line segment is selected in a biased 
random fashion between direction limits inclined at -45° 
and 0°, 

6 = -45° + 45° R 2 {?5) 

The same judgement, as discussed for critical shear 
surface searching with the circular and irregular-shaped 
shear surface generators, can also be applied to this 



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u 




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101 



generator, with regard to the probable existence of a 
significantly more critical shear surface than those 
generated. 



102 



COMPARISON OF CARTER'S METHOD WITH SPENCER'S 
METHOD ASSUMING PARALLEL SIDE FORCES 

A comparison was performed to assess the magnitude of 
error resulting from use of the AX=0 assumption. The 
comparison is of limited value, but it does demonstrate the 
nature of differences in values of the factor of safety to 
be expected. The Spencer method assuming parallel side 
forces was used as a standard. Even though the method as- 
suming parallel side forces does not, in general, produce a 
reasonable line of effective thrust, it does, however, 
satisfy the equilibrium conditions. Spencer (1973) has 
shown that solutions having an acceptable line of effective 
thrust do not yield values for the factor of safety signifi- 
cantly differently from those obtained when assuming parallel 
side forces. 

Combinations of the available shear strength intercept, 
c;, the available shear strength angle, <f^, the homogeneous 
pore pressure parameter, r u , and the slope inclination, 8, 
resulting in values for the factor of safety of unity, were 
selected from charts prepared by Spencer (1967). a height 
of slope equal to 100 ft and a moist unit weight for the 
soil of 100 pcf were assumed. Spencer's charts are based 
upon the critical circular shear surface passing through the 
toe of simple slopes composed of homogeneous 



103 

isotropic material and having a horizontal crest. Using 
the selected combinations of geometry and soil parameters, 
.values of the factor of safety (Table 2) were calculated 
for the critical surfaces using Carter's method. 

Curves fitted to the data found in Table 2 for the 
1.5:1 and 2:1 slopes are shown in Figure 30. Each curve 
represents the variation of percent difference of the value 
of the factor of safety found while comparing Carter's 
method to Spencer's, as the combination of the shear strength 
parameters c^ and $' vary. The curves are plotted with re- 
spect to <j>^, but it is implied that as the magnitude of *' 
increases, the value of c^ decreases in a complimentary 
manner to maintain a value of the factor of safety of unity, 
based upon Spencer's charts. 

When <f>'=0, the value of c* required to maintain a 
slope with a factor of safety of unity is identical for the 
two methods. For this case, AX for each slice does not 
contribute frictional resistance to the resisting shear 
force at the base of each slice. When c'=0, the value of 
<P' a required to maintain a slope with a factor of safety of 
unity is also identical for the two methods. For this case, 
the critical shear surface is at the face of the slope. AX 
is equal to zero for each slice, and the side forces are 
parallel, thus the agreement between the two methods. 
Differences occur when both c' and $ ' are non-zero. 



a T a 



104 









Table 2 








Factors 


of Safety Calculated Using Carter's Method* 


Slope 


♦; 

(deg) 


r = 
u 


0.5 


r u = °" 


,25 


r = 
u 







c;( P s 


f) F 


c^(psf) 


F 


c* (psf) 


F 




40 


450 


.864 


70 


.959 


_-. 


__ 


1.5:1 


30 


700 


.885 


290 


.917 


45 


.976 




20 


950 


.907 


610 


.916 


345 


.939 




10 


1250 


.958 


1040 


.955 


845 


.950 




40 


175 


.903 


_ _ 


_ _ 






2:1 


30 


425 


.892 


85 


.955 


— 


-- 




20 


725 


.892 


385 


.915 


140 


.943 




10 


1075 


.919 


850 


.917 


650 


.923 




40 


— 


— 


— 


_ _ 


mi _, 




3:1 


30 


100 


.947 


— 


-- 


— 


— 




20 


400 


.904 


110 


.973 


— 


— 




10 


850 


.897 


580 


.900 


370 


.920 




40 


— 


« 


— 


__ 


, m m 




4:1 


30 


— 


-- 


— 


— 


— 


— 




20 


190 


.942 


— 


— 


-- 


— 




10 


680 


.903 


380 


.910 


180 


.951 



Height of each slope equal to 10 ft, and moist unit 
weight equal to 100 pcf . 



105 




l-d poqjaw sjasuads uioj; aouaj^JQ ju«3ja d 



CD 



106 



If <j>^, for a slope having a factor of safety of unity 
when c; = 0, is reduced, while c^ is increased to maintain 
a factor of safety of unity, the critical shear surface will 
curve (Figure 31). When this happens, AX for each slice 
becomes non-zero, and an error is introduced. As the value 
of *; approaches zero, the frictional resistance induced by 
AX of each slice becomes insignificant. 

Considering the uncertainties involved in slope sta- 
bility analysis, the error which can be expected, generally 
less than 10%, is not unreasonable. Consideration of a 
more accurate model is more an economic question of with 
respect to the additional computer time and effort. 



107 




On- 



O 

to 

_ i 
o 

u_ 
o 

Q 
UJ 
(O 
O 
Q- 

o 
(J 

UJ 

o_ 

3 

(f) 
U. 

o 

LU 

or 

z> 

5 



to 

W 

or 
u. 



108 



SUMMARY AND RECOMMENDATIONS 
FOR FUTURE WORK 

The primary objective of this research was the develop- 
ment of a computer program capable of handling the general 
slope stability problem. The resulting program can analyse 
slope profiles having multiple slope ground surfaces, and 
containing any arrangement of subsurface soil types having 
differing soil properties. Pore pressure may be related to 
a steady state flow domain, related to the overburden, or 
specified within zones. Uniform boundary surcharge loads 
can be specified upon the ground surface. Pseudo-static 
earthquake loads, having vertical and horizontal components, 
are related to the weight of the mass above the assumed 
shear surface. 

Trial shear surface generators have been developed 
enabling random generation of surfaces of circular, general, 
or specified character. Each generator provides means for 
generation of only shear surfaces of particular interest. 

The program STABL, written in Fortran IV source 
language, is routinely used on a Control Data Corporation 
6500 computer at Purdue University. With full plotting 
capabilities 66K octal of core storage is required for 
execution. Program length is 6000+ statements including 
comments . 



109 



Although the program is very versatile, and efficient 
with respect to computing time, it is rather bulky. For 
-any problems only a fraction of its capability will be 
needed. The entire program must be loaded into a computer's 
memory core, denying possible use of a large portion of the 
oore to other programs. It is therefore suggested that a 
number of smaller, more specialized, and simpler programs 
be developed from STABL for the more efficient solution of 
some practical problems. 

Comparisons with solutions satisfying compiete equili- 
brium, and requiring reasonable lines of effective thrust 
and interslice shear stress, should be made to more ac- 
curately assess the importance of t-he =<^i. * 

P itam.e or the side force assumption. 

The method of slices i <. a „~^.. 

ces ls a ver y versatile limiting 

equilibrium model. It has potential applicatlons ^ ^ 

slope stability analysis. for example, this work could also 

be extended to laf-prai Q = >-4-v, 

to lateral earth pressure problems, retaining 

wall design, and bearing capacity analysis. 

To investigate the impact of input variables, such as 
soil properties, geometry, pore pressure distributions 
boundary loads, and earthquake loadings, on the solution of 
slope stability problems, a parametric or sensitivity study 
should be performed on STABL. Information resulting from 
such a study would enable identification of elements of a 
problem which require special attention. 



LIST OF REFERENCES 



110 



LIST OF REFERENCES 

Bishop, A. w. (1955) "rvu „ 

Stability Analysis of Slopes "V^ ^ CirCle in *he 
No. 1, March, Pp . 7 _ 17 ^° pes ' Ceotechnique^ Vol. 5, 

N °- 4, Dec, pp. f?9-J 5 PSS ' =2HtS£hni3ue, V ol? 10 _ 

Boussinesq, j (lsftS) « 

I' Etude de'l-Equiiibre^^rM 10 " deS Pote ntiels a 
Elastiques," Gauthier-vlllfrs plrTr" deS S ° lideS 

Carter, R. K . (1971) ..- 

Analysis by Method of ^f^" ^^ S1 °P e Stability 
University, West Laf aye ttl" {^^^^s , Purdue 

Coulomb, c. a. (1776) "p^ • 

Regies des Maxi mis *et nl nf™ i ^ U " e Application des 
Statique Relates aMrchTi 3 ? Quel <3 ues Problemes de 

!te =2i=£SLSai^sa4lS Histoir^et 

Fpllo . " ^^^^S^iences, Vol. 7, Paris. 

Fellenius, w. (1927) »p^ 4. • 

Reibung und Kohasion," wJlhelm^ 6 B f rich ™ngen «nlt 
Berlin. ' Wj -lnelm Ernst and Sohn KG, 

Frohlich, o. K. (1953) "n. =■ 

to Sliding of a Mass of Soif^ ° f Safet ^ With Respect 
rxthmic Spiral," Proceedings oTL^L^ ° f a ^ga- 
Conference on SoiT~M^^~' 5 he Thlrd International 
Switzerland, Vol. 2^^%^/^^^ ^—^ 

Frohlich, o. K. (1955) «r a 

S1 °Pes," C^o^hnijie, Vo? T N^TV" Stabil ity of 

"*— ' 3 ' NO ' *• March, pp. 37-54. 

Hansen, j. B . (I953) P=5 ., „ 

Technical Pre S s/iopih1fin^ £H£ ^2lculatio il , Danish 

Harr, m. e mokci 1-, 

panics.: ^^^silafegJaaarstissJjaii 

Janbu, n. (1954) "q^k- 1 • 

Dimensionless" Par Meters 1 '" j!^? ° f S1 °P es wi th 

N °- 46, January. meters ' ^£^£d_^oUjMech^m L c^_^erie s , 



Ill 

Janbu, N. (1957). " Earfh p 

Calculations by General f ^ res and Be «mg Capacitv 
^ceedi^, / th G f^-^d Procedure of Slfces^^ 

Soxl Mechanics and Foundatior £a£°^ Conf ™ on 
VOJ-. 2, pp. 207-212. engineering, London, 

Janbu, N. (1973) » S i 

^^Da iL _Engj^ee r inq P V t ? bi i ity Com Putations , » Emb ank 
ia77 7 iii ^^e-VoTu^ e John H " r f chfeld and S.'j.^gj 
PP. 47-86. Ume ' John Wiley & sons, New York, 

Morgenstern, n. r fln ^ D • 

of the Stability ol General V' U965) ' "^e Analysis 
S^t^iHLiaue, vol. 15 mo i m P Surf ^es,» Anai ysis 

' N °- X ' March > PP- 79-93. 
Morgenstern, n r *„,* n • 

"ethod for So^inTthe 1 !^^:^ <»«> . - A Numerical 

Nonveiller e MQfi^ 

«th a sup i ur 9 f 6 f^ of^ene'ral^ A ^ lysis °* Slopes 
the Sixth International rrnf Sna Pe." Proceedings of 
foundation Engineer™^ SSSSS^vS S f riiS ^cs™* 
Alston, A. and Wllf s . ', ' ' PP - 522 " 525 - 

^^^ital^p^; ^»l^- t i^ati.^jethods 

Romani F ., Lovell , Jr 

Influence of Progress!^ pL ■ ? Harr ' M " E - (1972) 
£§H£nal of the Soil MechJnf^ Ur ! ° n Slo P e Stability - 

Seed T V01 ' 98 ' N °" ^-'^P ^9- a i22°3 n r ^'^ 

^I.L?i^^ S ^^^~^^if f7 Jou«f ^i X± ^ Aaaly-e- 

i^ecnanics and Foundations ni ,, c -™^ of the Soil 
SM4 < July, p p . 6 9-83 Dl v 1S ion, ASCE, Vol. 93, No. 

Siegel, r. a . (lg? 
Spencer, e (1967) 



112 



Spencer, E. (1973) . "Thrust Line Criterion in Embankment 
Stability Analysis," Geotechnigue , Vol. 23, No. 1, 
March, pp. 85-100. 

Taylor, D. W. (1937). "Stability of Earth Slopes," Journal 
of the Boston Society of Civil Engineers, Vol. 24, No. 
3, July. 

Whitman, R. V. and Baily, W. A. (1967) . "Use of Computers 
for Slope Stability Analysis," Journal of the Soil 
Mechanics and Foundations Division, ASCE, Vol. 93, 
No. SM4, July, 475-498. 

Whitman, R. V. and Moore, P. J. (1963) . "Thoughts Concerning 
the Mechanics of Slope Stability Analysis," Proceedings , 
of the Second PanAmerican Conference on Soil Mechanics 
and Foundation Engineering, Brazil, Vol. 1, pp. 391-411. 

Wright, S. G. (1969). "A Study of Slope Stability and the 
Undrained Shear Strength of Clay Shales," Ph.D. Thesis , 
University of California, at Berkeley, California. 

Wright, S. G. , Kulhawy, F. H., and Duncan, J. M. (1973). 
"Accuracy of Equilibrium Slope Stability Analysis," 
Journal of the Soil Mechanics and Foundations Division, 
ASCE, Vol. 99, No. SM10, Oct., pp. 783-791. 



APPENDIX A 



LISTING OF PROGRAM STABL 



This Appendix is not included in this copy of this Report 
because of its size (98 pages) . Any person receiving a 
copy of this Report will be provided a copy of the Listing 
without charge upon written request to the Joint Highway 
Research Project, Civil Engineering Building, Purdue 
University, West Lafayette, Indiana, 47907. The request 
should specify the "Listing of Program Stabl" for Report 
JHRP-75-8. The several subroutines of the Program are 
listed in the Table of Contents of this Report. 



STABL USER MANUAL 

A STABL USER MANUAL has also been prepared and is 
available as a separate publication with title as above, 
Publication JHRP-75-9, June 1975. Availability is from 
the Joint Highway Project, address as above. The cost 
is $5/copy. 



g